You and your friend each do some experiment in which the sta

You and your friend each do some experiment in which the standard normal probability model is appropriate. X is your result, and Y is your friend\'s. Assume X and Y are independent.

a. What is the probability that X and Y are both less than 1?

b. What is the probability that one or both of X and Y are less than 1?

c. What is the probability that exactly one of X and Y is less than 1?

d. What is the conditional probability that X is less than 1, given that Y is?

Solution

A)

Using a table/technology,

P(X<1) = P(Y<1) = 0.841344746

Hence,

P(both less than 1) = P(X<1)P(Y<1) = 0.707860982 [ANSWER]

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B)

P(one or both are less than 1) = 1 - P(both are not less than 1 = 1 - (1-0.841344746)(1-0.841344746) = 0.97482851 [answer]

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c)

P(exactly one) = [P(X<1) - P(X<1)P(Y<1)] + [P(Y<1)-P(X<1)P(Y<1)] = 0.266967529 [ANSWER]

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D)

They are independent. Thus,

P(X<1|y<1) = P(X<1) = 0.841344746 [ANSWER]