Explanation Given: The probability that tennis player will serve an ace is 0.15 . Number of serves is 5 . Approach: The expanded expression of C ( n , x ) is given as; C ( n , x ) = n ! x ! ( n − x ) ! If p is the probability of success of an event and q is the probability of failure of an event. Then, the relationship between p and q is given as: q = 1 − p The probability of x successes in n independent trials is given by, P ( X = x ) C ( n , x ) p x q n − x Such that, x = 0 , 1 , 2 , 3 ......... , n Here, X is the binomial random variable and the probability distribution of X is called a binomial distribution. Calculation: Let X denote the number of times the tennis player serves an ace. The number of serves is n = 5 . The probability of serving an ace is, p = 0.15 Calculate the value of q . Substitute 0.15 for p in the expression for q . q = 1 − 0.15 = 0.85 Write the expression to calculate the probability of at least 2 serves. P ( X ≥ 2 ) = P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 ) + P ( X = 5 ) …… ( 1 ) Write the binomial distribution expression to calculate the probability of success of an event. P ( X = x ) = C ( n , x ) p x q n − x = n ! x ! ( n − x ) ! p x q n − x …… ( 2 ) Calculate the value of P ( X = 2 ) . Substitute 5 for n , 2 for x , 0.15 for p and 0.85 for q in equation ( 2 ) . P ( X = 2 ) = 5 ! 2 ! ( 5 − 2 ) ! ( 0.15 ) 2 ( 0.85 ) 5 − 2 = ( 120 2 × 6 ) ( 0.0225 ) ( 0.614125 ) = 0.1382 Calculate the value of P ( X = 3 ) . Substitute 5 for n , 3 for x , 0

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Author: Tan

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Chapter 8.4, Problem 23E

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The probability that a tennis player will serve at least two aces out of five serves.

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The probability that a tennis player will serve at least two aces out of five serves.

Solution Summary: The author explains how the probability of a tennis player serving an ace is 0.165.

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