1. Consider a sample space S and three events A, B, and C. For each of the follo
ID: 3353081 • Letter: 1
Question
1. Consider a sample space S and three events A, B, and C. For each of the following events draw a Venn diagram representation as well as a set expression. a. Among A, B, and C, only A occurs. b. At least one of the events A, B, or C occurs. c. A or C occurs, but not B. d. At most two of the events A, B, or C occur. 2. Let A, B, and C be three events in the sample space S. Suppose we know · PCA 12, P(B)-23, P(AUB)=56. · Answer the following questions: Find P(AnB). .Do A, B, and C form a partition of S? Find P(C-(AUB). If P(Cn(AUB))=512, find P(C). . 3. You toss a fair coin three times: a. What is the probability of three heads, HHH? b. What is the probability that you observe exactly one heads? c. Given that you have observed at least one heads, what is the probability that you observe at least two heads? 4. A box contains three coins: two regular coins and one fake two-headed coin (P(H)-1), You pick a coin at random and toss it. What is the probability that it lands heads up? You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin?Explanation / Answer
3)
X= number of heads
X follow binomial distribution with n = 3 , p = 1/2
a) P(X = 3) = (1/2)^3 = 1/8
b) P(X = 1) = 3C1 * (1/2)^3 = 3/8
c)P(X>= 2 |X>=1) = P(X>= 2 and X >= 1)/P(X >= 1)
= P(X>= 2 )/P(X >= 1)
now
P(X>=2) = P(X = 2) + P(X = 3) = 3/8 + 1/8 = 1/2
P(X>= 1) = 1 - P(X = 0) = 7/8
hence
required probability = (1/2)/(7/8)
= 4/7
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