Question

A student possesses five coins, two of which are double-headed, one of which is double-tailed, and two of whichare normal. She shuts her eyes, picks a coin at random, and tosses it. What’s the probability that the lowerface of the coin is a tail?

2.Suppose that a probability space includes three events,A,B, andC, withP(A) =P(B) =P(C) = 0.4. IfAandBare mutually exclusive, andBandCare mutually exclusive, areAandCindependent?

3.Alicia, Brunhilda, and Chloë all decide to throw caution to the wind and attend a house party. Each of the(32)= 3sub pairs hang out at the party with probabilityp, or don’t with probability1−p. Assume that hangingout occurs independently. If any one has a disease during the party, they will spread it to all others they hangout with (and so they will spread it to those that they hang out with, etc.). If Alicia comes to the party with adisease, what is the probability that(a) Chloë contracts the disease, given that Brunhilda does not hang out with Chloë?(b) Chloë contracts the disease, given that Brunhilda does hang out with Chloë?(c) Chloë contracts the disease?

4. During the trial of Alfred Dreyfus in 1894, the prosecution argued on the assumption thatP(A|B) =P(B|A).Is this true? If so, explain why; if not, give a counterexample.

5.Some form of prophylaxis is said to be 95 percent effective at prevention during one month’s treatment. Ifsuccesses in prevention in different months are independent, what is the probability that the treatment will failduring the first fourteen months?

Answer 1

Since no outcome of coin toss is mentioned, we just need to find the probability if the randomly selected coin has tail on one particular side. As per the cases,

Events are mutually exclusive if they cannot occur together, and in that case probability of them occuring together is zero. Events are independent if the outcome of one does not affect the other, and in that case the multiplication rule applies.

P(A | B) denotes the probability that A occurs, given that B has happened. P(B | A) denotes the probability that B occurs, given that A has happened. These probabilities are not necessarily the same. Theoretically,

For ex, in the random exeriment of tossing a coin 5 times, consider event A that at least one head was obtained and consider event B that at least two heads were obtained. It is pretty local that P(B | A) = 1, since if at least two head were obtained then at least one was definitely obtained. By a similar argument, P(A | B) is definitely not 1, since getting at least one head does not guarantee that at least two were obtained

Failure occurs if even in any one of the fourteen months, failure occurs. Since failures across various months are independent.

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