# conditional probability problems

Thus, it is useful to draw Let $$B=$$ the event the collector buys, and $$G=$$ the event the painting is original. Event A is the main focus: we are interested whether or not A occurs. probability more interesting. Suppose that a good part fails within the ﬁrst year with probability 0.01, while a sli ghtly defective part fails within the ﬁrst year with probability 0 … a. Here we have four possibilities, $GG=(\textrm{girl, girl}), GB, BG, BB$, Thus, we can use the law of total probability to write, Now, for the second part of the problem, we are interested in $P(C_2|H)$. Now we can write immediately obtain $c=\frac{1}{2}$, which then gives $a=\frac{1}{3}$ and $b=\frac{1}{2}$. P(A and B) is the probability of the occurrence of both A and B at the same time. In part (b) of Example 1.18, In fact, we are using CONDITIONAL PROBABILITY WORD PROBLEMS WORKSHEET. Solution. Let’s get to it! Independent Events . 25% of the class passed both tests and 42% of the class passed the first test. Let us now concentrate on the more complex conditional probability problems we began looking at above. The conditional probability table below shows how 300 cases, in all of which the player initially chooses door 1, would be split up, on average, according to the location of the car and the choice of door to open by the host. Consider $$P(C) = P(C^c) = 0.5$$, $$P(A|C) = 1/4$$. Understand conditional probability with the use of Monty Hall Problem. As it is seen from the problem statement, we are given conditional probabilities in a chain format. sequences that result in my winning. This is surprising to What is the (conditional) probability that the first and third selected are women, given that three of those selected are women? Suppose a person has a university education (no graduate study). We can find $P(R|L)$ using $P(R|L)=\frac{P(R \cap L)}{P(L)}$. Let $$T$$ = event test indicates defective, $$D$$ = event initially defective, and $$G =$$ event unit purchased is good. P(A or B) is the probability of the occurrence of atleast one of the events. The probability that it's not raining and there is heavy traffic and I am not late can What is the probability that three of those selected are women? In other words, we want to find I am giving two extra days on this problem set since I may not get to the Radon-Nikodym theorem before Thursday Oct 30. understanding of probability. Here is another variation of the family-with-two-children is $0.25$. Show that $$P(A|B) \ge (P(A) + P(B) - 1)/P(B)$$. Two percent of the units received at a warehouse are defective. one third in this case. Thus, it is useful to draw a tree diagram. d. $$P(B_1|R_2)$$, b. sample space here still is $GG, GB, BG$, but the point here is that these are not equally likely This is again similar to the previous problem (please read the explanation there). Let $$A_6$$ = event first is a six and $$S_k =$$ event the sum is $$k$$. Four persons are to be selected from a group of 12 people, 7 of whom are women. In this figure, each leaf in the tree corresponds to a single outcome in the sample space. traffic with probability $\frac{1}{2}$, and given that it is not rainy, there will be heavy Result of testimony: $$P(L|G)/P(L|G^c) = 6$$. I win if late for work with probability $\frac{1}{2}$. If you do not know the CDF of a geometric distribution you can do the following reasoning...the probability to have more than 5 failures is exactly the probability of having 5 consecutive failures...after this events any event can happen....thus you probability is. the assumptions about independence and disjointness of sets are already included in the figure. It is cleaner if we divide $W$ into two parts depending Okay, another family-with-two-children problem. There is an equally likely probability that there are 0, 1, 2, or 3 defective units in the lot. $$E_1$$= event did not complete college education; $$E_2$$= event of completion of bachelor's degree; $$E_3$$= event of completion of graduate or professional degree program. In this class we will treat Bayes' problems as another conditional probability and not involve the large messy formula given in the text (and every other text). Given that it is rainy, there will be heavy What is the (conditional) probability that one turns up two spots, given they show different numbers? For three events $A$, $B$, and $C$, we know that. We suppose $$P(B) = p$$ and $$P(A|B^c) = 1/n$$. Let $C_1, C_2,\cdots,C_M$ be a partition of the sample space $S$, and $A$ and $B$ be two events. Conditional probability and independence. The conditional sample space here still is $GG, GB, the probabilities of the outcomes that correspond to me being late. Also, let$B$be the event problem  . $$P(A^c|B) > P(A^c)$$ iff $$P(A^c B) > P(A^c) P(B)$$ iff $$P(AB) < P(A) P(B)$$ iff $$P(A|B) < P(A)$$, c. $$P(A|B) > P(A)$$ iff $$P(AB) > P(A) P(B)$$ iff $$P(A^c B^c) > P(A^c) P(B^c)$$ iff $$P(A^c|B^c) > P(A^c)$$. If it's rainy and there is heavy traffic, I arrive If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. $$P(E_3S_3) = P(S_3|E_3)P(E_3) = 0.45 \cdot 0.05 = 0.0225$$, b. visualize the events in this problem. \frac{1}{3}$, $= \frac{P(L|GG)P(GG)}{P(L|GG)P(GG)+P(L|GB)P(GB)+P(L|BG)P(BG)+P(L|BB)P(BB)}$, $= \frac{(2 \alpha-\alpha^2)\frac{1}{4}}{(2 \alpha-\alpha^2)\frac{1}{4}+ \alpha \frac{1}{4}+ \alpha \frac{1}{4}+0.\frac{1}{4}}$. The conditional A quality control group is designing an automatic test procedure for compact disk players coming from a production line. $$P(A|B) = \dfrac{P(AB)}{P(B)} = \dfrac{P(ABC) + P(ABC^c)}{P(B)}$$, $$= \dfrac{P(A|BC) P(BC) + P(A|BC^c) P(BC^c)}{P(B)} = P(A|BC) P(C|B) + P(A|BC^c) P(C^c|B)$$. The probability that a randomly chosen child \frac{2}{3} + 1 . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. A table of sums shows $$P(A_6S_k) = 1/36$$ and $$P(S_k) = 6/36, 5/36, 4/36, 3/36, 2/36, 1/36$$ for $$k = 7$$ through 12, respectively. The converse is not true. Two standard dice with 6 sides are thrown and the faces are recorded. Thus, the sample space reduces to three $$P(L|BG)=P(L|GB)=\alpha,$$ A conditional probability Pr(B | A) is called an a posteriori if event B precedes event A in time. Then we have You pick a coin at random and toss it. There are $$n$$ balls on the first choice, $$n + c$$ balls on the second choice, etc. Compound probability is when the problem statement asks for the likelihood of the occurrence of more than one outcome. Examples on how to calculate conditional probabilities of dependent events, What is Conditional Probability, Formula for Conditional Probability, How to find the Conditional Probability from a word problem, How to use real world examples to explain conditional probability, with video lessons, examples and step-by-step solutions. $$1 \ge P(A \cup B) = P(A) + P(B) - P(AB) = P(A) + P(B) - P(A|B) P(B)$$. A unit is examined and found to have the characteristic symptom. Let $C_1$ be the event that you choose a regular coin, and let $C_2$ be the event Experience shows that 93 percent of the units with this defect exhibit a certain behavioral characteristic, while only two percent of the units which do not have this defect exhibit that characteristic. b. Here again, we have four Conditional probability is the probability of an event occurring given that another event has already occurred. Second, after obtaining counterintuitive results, you are encouraged to think deeply Data on incomes and salary ranges for a certain population are analyzed as follows. $$P(B_1B_2)$$ Let’s take a real-life example. Example $$\PageIndex{7}$$ Suppose a certain disease has an incidence rate of 0.1% (that is, it afflicts 0.1% of the population). probability problems, probability, probability examples, how to solve probability word problems, probability based on area, How to use permutations and combinations to solve probability problems, How to find the probability of of simple events, multiple independent events, a union of two events, with video lessons, examples and step-by-step solutions. I purchase the product and use it for two years without that you choose the two-headed coin. probability of $GG$ is higher. Well, this … Determine $$P(A_i|B_0)$$ for each possible $$i$$. P(A or B) = P(A) + P(B) – P(A and B) where A and B are any two events. Just kidding! only one girl is $\frac{1}{2}$. What is the (conditional) probability that at least one turns up six, given that the sum is $$k$$, for each $$k$$ from two through 12? What is the probability that both children are girls? Show that if $$P(A|C) > P(B|C)$$ and $$P(A|C^c) > P(B|C^c)$$, then $$P(A) > P(B)$$. If one is selected at random and found to be good, what is the probability of no defective units in the lot? extra information about the name of the child increases the conditional probability of The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. In the last lesson, the notation for conditional probability was used in the statement of Multiplication Rule 2. Then, $$1/2 = P(A) = \dfrac{1}{2} (1/4 + 3/4) > \dfrac{1}{2} (1/2 + 1/4) = P(B) = 3/8$$. That is, if $$D$$ is the event a unit tested is defective, and $$T$$ is the event that it tests satisfactory, then $$P(T|D) = 0.05$$ and $$P(T^c|D^c) = 0.02$$. Conditional Probability Definition We use a simple example to explain conditional probabilities. $$P(D_0|G) = \dfrac{P(G|D_0) P(D_0)}{P(G|D_0) P(D_0) + P(G|D_1) P(D_1) + P(G|D_2) P(D_2) + P(G|D_3) P(D_3)}$$, $$= \dfrac{1 \cdot 1/4}{(1/4)(1 + 999/1000 + 998/1000 + 997/1000)} = \dfrac{1000}{3994}$$. These can be tackled using tools like Bayes' Theorem, the principle of inclusion and exclusion, and the notion of independence. What is the probability that you observe exactly one heads?