4.62 Find and explain some probabilities. (a) Can we have an event A that has ne

Question

4.62 Find and explain some probabilities. (a) Can we have an event A that has negative probability? Explain your answer. (b) Suppose P(A) = 0.2 and P(B) = 0.4. Explain what it means for A and B to be disjoint. Assuming that they are disjoint, find the probability that A or B occurs. (c) Explain in your own words the meaning of the rule P(S) = 1. (d) Consider an event A. What is the name for the event that A does not occur? If P(A) = 0.3, what is the probability that A does not occur? (e) Suppose that A and B are independent and that P(A) 0.2 and P(B) = 0.5. Explain the meaning of the event and B), and find its probability.

Explanation / Answer

Answer to the question is as follows:

a. No. there cannot be negative probabilities. So, event A cannot have
negative probability

b. If A and B is disjoint then the two sets have no element in common.
So, P(A or B) = P(A)+P(B) - P(A and B). Since A and B are disjoint P(A and B) = 0,
therefore, P(A or B) = P(A)+P(B) -0 = .2+.4 = .6

c. P(S) = 1, means that the probability of getting any of elements in sample space.
That would definite come to 1, as S is a superset of all sub sets.

d. A is an event, Ac is the event that A doesn't occur. P(A) = .3,
A doesn't occur with probability = P(Ac) = 1- P(A) = 1-.3 = .7

e. P(A and B) = Probability that both events happen = P(A and B) = P(A)*P(B). This conversion happens only when A and B are independent.
So, P(A and B) = P(A)*P(B) = .2*.5 = .10

When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring.

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