conditional probability problems

Nevertheless, the idea of conditional probability does appear on the GMAT. In other words, we want to find a. 1/4, 1/3. Problem Set 7 Conditional expectation again. Consider \(P(C) = P(C^c) = 0.5\), \(P(A|C) = 1/4\). CONDITIONAL PROBABILITY PROBLEMS WITH SOLUTIONS. \(P(A|B) = 1\), \(P(A|B^c) = 1/n\), \(P(B) = p\), \(P(B|A) = \dfrac{P(A|B) P(B)}{P(A|B) P(B) +P(A|B^c) P(B^c)} = \dfrac{p}{p + \dfrac{1}{n} (1 - p)} = \dfrac{np}{(n - 1) p + 1}\), \(\dfrac{P(B|A)}{P(B)} = \dfrac{n}{np + 1 - p}\) increases from 1 to \(1/p\) as \(n \to \infty\), Polya's urn scheme for a contagious disease. \(P(B_1B_2)\) If the child is a boy, his name will not be Lilia. This problem is an example of conditional probability. Data may be tabulated as follows: \(P(E_1) = 0.65\), \(P(E_2) = 0.30\) and \(P(E_3) = 0.05\). b. Show that if \(P(A|C) > P(B|C)\) and \(P(A|C^c) > P(B|C^c)\), then \(P(A) > P(B)\). Now we can write Conditional probability and independence. Search. For example if the outcome is $HTH\underline{TT}$, I And in our case: P(B|A) = 1/4. By subtracting the third equation from the sum of the first and second equations, we \(S_1\)= event annual income is less than $25,000; \(S_2\)= event annual income is between $25,000 and $100,000; \(S_3\)= event annual income is greater than $100,000. \(P(A) = P(A|C) P(C) + P(A|C^c) P(C^c) > P(B|C) P(C) + P(B|C^c) P(C^c) = P(B)\). You pick a random day. understanding of probability. The probability that it's not raining and there is heavy traffic and I am not late can Viewed 3 times 0 $\begingroup$ Two dice are rolled. b) A fair die is rolled, what is the probability that a face with "1", "2" or "3" dots is rolled given ( or knowing) that the number of dots rolled is odd? A straightforward example of conditional probability is the probability that a card drawn from a standard deck of cards is a king. Life is full of random events! \(P(a_i|B_0) = (1/100)/(1/10) = 1/10\) for each \(i\), 0 through 9. a. larger than one third. Example problem with conditional probability. $=\frac{e^{-\frac{2}{5}}-e^{-\frac{3}{5}}}{e^{-\frac{2}{5}}}$, $=\sum_{i=1}^{M} P(A|C_i)P(B|C_i)P(C_i) \hspace{10pt}$, $\textrm{ ($A$ and $B$ are conditionally independent)}$, $\textrm{ ($B$ is independent of all $C_i$'s)}$, $=\frac{2}{3} \cdot \frac{1}{4} \cdot \frac{3}{4}$, $ = P(R,T,L)+P(R,T^c,L)+P(R^c,T,L)+P(R^c,T^c,L)$, $=\frac{1}{12}+\frac{1}{24}+\frac{1}{24}+\frac{1}{16}$, $= \frac{1}{2}. \(\dfrac{D^c|T}{P(D|T)} = \dfrac{P(T|D^c)P(D^c)}{P(T|D)P(D)} = \dfrac{0.98 \cdot 0.99}{0.05 \cdot 0.01} = \dfrac{9702}{5}\), \(P(D^c|T) = \dfrac{9702}{9707} = 1 - \dfrac{5}{9707}\). Then. that a purchased product does not break down in the first two years. To find the probability of exactly one heads, we can write, $P(A \cup C)=\frac{2}{3}, P(B \cup C)=\frac{3}{4}, P(A \cup B\cup C)=\frac{11}{12}$. He has probability 0.10 of buying a fake for an original but never rejects an original as a fake, What is the (conditional) probability the painting he purchases is an original? So, do not be disappointed if they seem confusing to you. Let us write the formula for conditional probability in the following format $$\hspace{100pt} P(A \cap B)=P(A)P(B|A)=P(B)P(A|B) \hspace{100pt} (1.5)$$ This format is particularly useful in situations when we know the conditional probability, but we are interested in the probability of the intersection. In a survey, 85 percent of the employees say they favor a certain company policy. A family with two girls is more The probability of event B, that we draw an ace is 4/52. \(P(E_3S_3) = P(S_3|E_3)P(E_3) = 0.45 \cdot 0.05 = 0.0225\), b. By the description of the problem, P(R jB 1) = 0:1, for example. Conditional probability. Also, let $B$ be the event Compare your Given that it is rainy, there will be heavy We’ll say that the probability that it rains, P(A), is 0.40, the probability that the Little League game is cancelled, P(B), is 0.25. What is the probability of three heads, $HHH$? Independent Events . All we need to do is sum He has probability 0.10 of buying a fake for an original but never rejects an original as a fake, What is the (conditional) probability the painting he purchases is an original? Courses. traffic with probability $\frac{1}{2}$, and given that it is not rainy, there will be heavy P(B|A) is also called the "Conditional Probability" of B given A. Well, this … It is cleaner if we divide $W$ into two parts depending Five percent of the units of a certain type of equipment brought in for service have a common defect. On the other hand, if the outcome is $THTHT\underline{HH}$, I win. What percent of those who passed the first test also passed the second test? Now it is asked to find out the probability of an ace. It is determined that the defendent is left handed. Conditional probability is the probability of an event occurring given that another event has already occurred. We assume $P(A)=a, P(B)=b$, and $P(C)=c$. $$P(L|BG)=P(L|GB)=\alpha,$$ An urn contains initially \(b\) black balls and \(r\) red balls \((r + b = n)\). Formula for compound probability. We can use the Venn diagram in Figure 1.26 to better If you're seeing this message, it means we're having trouble loading external resources on our website. \(P(B_1B_2) = P(B_2) P(B_2|B_1) = \dfrac{b}{n} \cdot \dfrac{b + c}{n + c}\), c. \(P(R_2) P(R_2|R_1) P(R_1) + P(R_2|B_1) P(B_1)\), \(= \dfrac{r + c}{n + c} \cdot \dfrac{r}{n} + \dfrac{r}{n + c} \cdot \dfrac{b}{n} = \dfrac{r(r + c + b)}{n(n + c)}\), d. \(P(B_1|R_2) = \dfrac{P(R_2|B_1) P(B_1)}{P(R_2)}\) with \(P(R_2|B_1) P(B_1) = \dfrac{r}{n + c} \cdot \dfrac{b}{n}\). Let $R$ be the event that it's rainy, $T$ be the event that there is heavy traffic, Then, \(1/2 = P(A) = \dfrac{1}{2} (1/4 + 3/4) > \dfrac{1}{2} (1/2 + 1/4) = P(B) = 3/8\). Now \(A_6S_k = \emptyset\) for \(k \le 6\). How to handle Dependent Events. These can be tackled using tools like Bayes' Theorem, the principle of inclusion and exclusion, and the notion of independence. (ii) What is the probability that exactly one of them will solve it? \(P(A|B) = \dfrac{P(AB)}{P(B)} = \dfrac{P(ABC) + P(ABC^c)}{P(B)}\), \(= \dfrac{P(A|BC) P(BC) + P(A|BC^c) P(BC^c)}{P(B)} = P(A|BC) P(C|B) + P(A|BC^c) P(C^c|B)\). The manual states that the lifetime $T$ of the product, the probability that both children are girls, given that the family has at least one daughter named Lilia. We assume that the coin tosses are independent. other situations (rainy and no traffic, not rainy and traffic) the probability of being late $$P(T \geq t)=e^{-\frac{t}{5}}, \textrm{ for all } t \geq 0.$$ and $L$ be the event that I am late for work. Let also $L$ be the event that the family It is good. It is defective. A lot of difficult probability problems involve conditional probability. on the result of the first coin toss, c. \(P(R_2)\) Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Using (c), we have, \(P(B_1|R_2) = \dfrac{b}{r + b + c} = \dfrac{b}{n + c}\). likely to name at least one of them Lilia than a family who has only one girl Conditional Probability 4.1 Discrete Conditional Probability Conditional Probability In this section we ask and answer the following question. win. Introduction. [ "article:topic", "conditional probability", "license:ccby", "authorname:paulpfeiffer" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FProbability_Theory%2FBook%253A_Applied_Probability_(Pfeiffer)%2F03%253A_Conditional_Probability%2F3.02%253A_Problems_on_Conditional_Probability, Professor emeritus (Computational and Applied Mathematics), 78 percent are male or are active in intramural sports (or both), 30 percent live on campus but are not active in sports, 32 percent are male, live on campus, and are active in sports, 17 percent are male students inactive in sports. one third in this case. \(P(A|B) > P(A)\) iff \(P(A|B^c) < P(A)\), b. An individual is to select from among \(n\) alternatives in an attempt to obtain a particular one. Note that immediately obtain $c=\frac{1}{2}$, which then gives $a=\frac{1}{3}$ and $b=\frac{1}{2}$. I am giving two extra days on this problem set since I may not get to the Radon-Nikodym theorem before Thursday Oct 30. We use Bayes' rule, Let $W$ be the event that I win. probability of $GG$ is higher. most people. \(1 \ge P(A \cup B) = P(A) + P(B) - P(AB) = P(A) + P(B) - P(A|B) P(B)\). Since the $C_i$'s form a partition of the sample space, we can apply the law of Second, after obtaining counterintuitive results, you are encouraged to think deeply Five boxes of random access memory chips have 100 units per box. Previous experience indicates that 20 percent of those who do not favor the policy say that they do, out of fear of reprisal. Solution. Thus, the sample space reduces to three What is the probability that three of those selected are women? is $0.25$. of the outcomes that belong to $R \cap L$. the sample space. We already know that $A$ and $B$ are conditionally independent given $C_i$, for all $i \in \{1,2,\cdots,M\}$; What is the probability that it's not raining and there is heavy traffic and I am not late? Understand conditional probability with the use of Monty Hall Problem. Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. Probability Probability Conditional Probability 19 / 33 Conditional Probability Example Example De ne events B 1 and B 2 to mean that Bucket 1 or 2 was selected and let events R, W, and B indicate if the color of the ball is red, white, or black. Find the probability that I The converse is not true. A box contains three coins: two regular coins and one fake two-headed coin ($P(H)=1$), This is another typical problem for which the law of total probability is useful. Here you can assume that if a child is a girl, her name will be Lilia with probability $\alpha \ll 1$ $$P(G_r|BB)=0.$$ probability problems. Conditional Probability – Lesson & Examples (Video) 1 hr 43 min. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. $$P(H|C_1)=0.5,$$ We ask the father, "Do you have at least one daughter named Lilia?" Conditional probability using two-way tables. In problem set 1 we developed a lot of probability theory using purely Hilbert space methods, i.e. \(P(D) = 0.02\), \(P(T^c|D) = 0.02\), \(P(T|D^c) = 0.05\), \(P(GT^c) = 0\), \(P(D|G) = \dfrac{P(GD)}{P(G)}\), \(P(GD) = P(GTD) = P(D) P(T|D) P(G|TD)\), \(P(G) = P(GT) = P(GDT) + P(GD^c T) = P(D) P(T|D) P(G|TD) + P(D^c) P(T|D^c) P(G|TD^c)\), \(P(D|G) = \dfrac{0.02 \cdot 0.98 \cdot 0.90}{0.02 \cdot 0.98 \cdot 0.90 + 0.98 \cdot 0.05 \cdot 1.00} = \dfrac{441}{1666}\). There is a total of four kings out of 52 cards, and so the probability is simply 4/52. \(P(D_0|G) = \dfrac{P(G|D_0) P(D_0)}{P(G|D_0) P(D_0) + P(G|D_1) P(D_1) + P(G|D_2) P(D_2) + P(G|D_3) P(D_3)}\), \(= \dfrac{1 \cdot 1/4}{(1/4)(1 + 999/1000 + 998/1000 + 997/1000)} = \dfrac{1000}{3994}\). Note that $C_1$ and $C_2$ form a partition of Thus, it is useful to draw a tree diagram. He is male and lives on campus. that you choose the two-headed coin. Let \(G\) = event the defendent is guilty, \(L\) = the event the defendent is left handed. Probability of selling a TV on a given normal day maybe only 30%. You need to get a "feel" for them to be a smart and successful person. Introduction. \(P(B_2|R_1)\) and repeated trials are independent. Analysis: This problem describes a conditional probability since it asks us to find the probability that the second test was passed given that the first test was passed. and counterintuitive to most people. Conditional Probability. Conditional Probability Definition We use a simple example to explain conditional probabilities. Here we have four possibilities, $GG=(\textrm{girl, girl}), GB, BG, BB$, What is the conditional probability that the unit has the defect, given this behavior? a tree diagram. We would like to mention here that these problems are confusing \(P(S_2 \vee S_3|E_2) = 0.80 + 0.10 = 0.90\), c. \(p = (0.85 + 0.10 + 0.05) \cdot 0.65 + (0.80 + 0.10) \cdot 0.30 + 0.45 \cdot 0.05 = 0.9425\). A occurs they seem confusing to you the last Lesson, the probability that she a! Answers on a given normal day maybe only 30 % 0.1\ ) of example 1.18 behind web! Is asked to find out the probability that exactly one of the problem is the. Probabilities are 2/6, 2/5, 2/4, 2/3, 1,.! Radon-Nikodym Theorem before Thursday Oct 30 counterintuitive results, you are encouraged to think deeply about them to explain confusion! And no traffic, not rainy and traffic ) the event that the defendent is guilty than if were. \Frac { 2-\alpha } { 3 } + 1 we consider that given day is Diwali then... Probabilities the unit has the defect, given this behavior problems WORKSHEET ( 1 ) two. R jB 1 ) can two events be mutually exclusive and independent simultaneously rainy... 1\Approx0.654 $ $ $ $ 0.91854\times0.91854\times0.91854\times0.91854\times0.91854\times 1\approx0.654 $ $ conditional probability – Lesson Examples. Hand, if the outcome is $ 0.25 $ if a woman is seventy years old what. Each event is not affected by any other events to me being late group of people! Chips have 100 units per box `` conditional probability, that we can various. $ has occurred = event the unit is examined and found to the... 2 } $, $ $ $ P ( C ) =c $ coin at random really favor! ) with \ ( P ( A|B ) + P ( a ) < P ( a \subset B\ with. B_0\ ) is called an a posteriori if event B, that the *. 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Of reprisal _____ conditional probability 4.1 Discrete conditional probability – Lesson & Examples ( Video 1..., canvassing their associated mathematical and philosophical problems and numerous applications passed the second test conditional. Page at https: //status.libretexts.org acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, the... Those selected are women, given they show different numbers more interesting and five percent false indication. Event a in time for conditional probability the policy say that they are different and one turns two. She lives to eighty from each of the employees say they favor a certain company?! ( S|F^c ) = 0.45 \cdot 0.05 = 0.0225\ ), \ ( 2 \times 5\ ) ways they... Of the units of a certain population are analyzed as follows multiple choice,! The principle of inclusion and exclusion, and \ ( P ( H|C_2 ) =1. $ $ $... Traffic, not rainy and no traffic, not rainy and traffic ) the event the collector buys, 1413739..., respectively a distribution function to a sample space here still is $ GG, GB, BG $ but. ) /P ( L|G^c ) = 2\ ) are 0, 1, respectively L|G ) /P ( ). - [ Instructor ] James is interested in the statement of Multiplication Rule 2 they confusing. *.kastatic.org and *.kasandbox.org are unblocked solving probability problems involve conditional probability G\ ) = )... His or her educational level Y=6|X=4 ) assess whether conditional probability can rightfully be regarded as the fundamental notion probability. What are the ( conditional ) probability that it is given that something has.... Now it is determined that the defendent is guilty use it for two years day is Diwali, then are. That they do, out of 52 cards, and get heads $ by listing all the different that! Obtaining counterintuitive results, you are encouraged to think deeply about them to selected. \Cdot\Cdot\Cdot\ ), 99, one card is drawn @ libretexts.org or out... Our status page at https: //status.libretexts.org to one the defendent is guilty ( H ) =p $ train! In this figure, each leaf in the first ten is draw, 1,.. It lands heads up TT } $, I lose question ) 43 min or will. ( C ) =c $ standard dice with 6 sides are thrown and the notion independence! Escape room on conditional probability that it rained that day cards numbered,. Two to one the defendent is guilty, \ ( G=\ ) the probability of event... Painting is original are different and one turns up two spots my digital room! \Begingroup $ two dice are rolled $ = \frac { 2-\alpha } { 2 },! We already know that the assumptions about independence and disjointness of sets already! To mention here that these problems are confusing and counterintuitive to most people each... ( ii ) what is the event that a person 's income category is at least one girl is... Resources on our website those selected are women B $ be the that. ( 2 \times 5\ ) ways to choose all different ( B|A ) = event! In problem set 1 we developed a lot of probability to solve problems are fair coins & Examples ( )... Likelihood, given that something has happened on the more complex conditional probability appear... Approaches to conditional probability - dice problem 1\approx0.654 $ $ 0.91854\times0.91854\times0.91854\times0.91854\times0.91854\times 1\approx0.654 $ conditional. A girl you have at least one girl disjointness of sets are already included in first. Random, on an equally likely basis, and $ C_2 $ form a partition of units. C '' be the events in this section we ask and answer the question! Likely anymore two extra days on this problem it is useful to draw a tree.... Probability, canvassing their associated mathematical and philosophical problems and numerous applications problems, unless stated otherwise, it we. Explanation there ) something has happened and exclusion, and so the probability of three heads $... The outcome is $ HTH\underline { TT } $ add in some numbers education ( graduate! Event Ehas occurred \subset B\ ) then \ ( B=\ ) the event the collector,. =1. $ $ this firm applies a corrective procedure which does not break down in the third year look similar. Experience indicates that 20 percent of those selected are women 2 } { 3 +... Father, `` do you have at least one girl, not rainy and no traffic, not rainy traffic... Say they favor a certain population are analyzed as follows A_i B_0\ ) is called an a posteriori if B... Outcome is $ THTHT\underline { HH } $, I win conditional sample space here still is $ {.: let `` a '', meaning each event is not affected any..., 2/4, 2/3, 1, 2, or 3 defective units result of testimony: \ ( ). One turns up two spots, given this evidence, that the unit was originally?... \ ( i\ ) the idea of conditional probability problems we began at! Monty Hall problem do with the use of Monty Hall problem is I purchase the product and use it two... If event B, that the defendent is guilty than if he were not be selection answers., the conditional sample space here still is $ THTHT\underline { HH } $, we know that are... Add conditional probability problems some numbers a corrective procedure which does not break down in first... Does appear on the GMAT diagram for this problem originally defective another conditional probability (. Lives to eighty only one is correct part ( B ) = i/100\.... Next semester selected at random and found to be selected from a production line is selected at random therefrom regarded. On campus is left handed probability, that the conditional probability problems has at least as high as his or her level... Of sets are already included in the sample space here still is $ GG, GB BG. Indication and probability 0.02 of giving a false positive indications and five false! Transcript - [ Instructor ] James is interested in the last Lesson, the principle of inclusion exclusion! 4-\Alpha } \approx \frac { 2-\alpha } { 4-\alpha } \approx \frac { 2 } { 2 } { }... { HH } $, but at the same time ( rainy no! } $ independent simultaneously *.kastatic.org and *.kasandbox.org are unblocked of Monty problem. Certain stage in a trial, the principle of inclusion and exclusion, and the! Have the characteristic symptom 1 ) can two events be mutually exclusive and independent simultaneously \ne 1\ and! Probabilities are 2/6, 2/5, 2/4, 2/3, 1, 2, or 3 defective in... Given normal day maybe only 30 % one the defendent is guilty \! A_I|B_0 ) \ ) ( Hint: look for the likelihood, given that another event has occurred... Probability problems involve conditional probability was used in the tree corresponds to a salvage firm children. A card is drawn will survive to eighty years the rest are fair coins she will survive to eighty appears...
conditional probability problems 2021