CHAP. 7] PROBABILITY 151
REPEATED TRIALS, BINOMIAL DISTRIBUTION
7.67. Whenever horsesa,b, andcrace together, their respective probabilities of winning are 0.3, 0.5, and 0.2. They race
three times.
(a) Find the probability that the same horse wins all three races.
(b) Find the probability thata,b,ceach win one race.
7.68. The batting average of a baseball player is 0.300. He comes to bat four times. Find the probability that he will get:
(a) exactly two hits; (b) at least one hit.
7.69. The probability that Tom scores on a three-point basketball shot isp= 0 .4. He shootsn=5 times. Find the
probability that he scores: (a) exactly two times; (b) at least once.
7.70.A certain type of missile hits its target with probabilityP=^13
(a) If three missiles are fired, find the probability that the target is hit at least once.
(b) Find the number of missiles that should be fired so that there is at least a 90% probability of hitting the target.RANDOM VARIABLES
7.71.A pair of dice is thrown. LetXdenote the minimum of the two numbers which occur. Find the distributions and
expectation ofX.
7.72.A fair coin is tossed four times. LetXdenote the longest string of heads. Find the distribution and expectation ofX.
7.73.A fair coin is tossed until a head or five tails occurs. Find the expected numberEof tosses of the coin.
7.74.A coin is weighted so thatP(H)=^34 andP(T)=^14. The coin is tossed three times. LetXdenote the number of
heads that appear.
(a) Find the distributionfofX. (b) Find the expectationE(X).
7.75. The probability of teamAwinning any game is^12. SupposeAplaysBin a tournament. The first team to win two games
in a row or three games wins the tournament. Find the expected number of games in the tournament.
7.76.A box contains 10 transistors of which two are defective. A transistor is selected from the box and tested until
a nondefective one is chosen. Find the expected number of transistors to be chosen.
7.77.A lottery with 500 tickets gives one prize of $100, three prizes of $50 each, and five prizes of $25 each.
(a) Find the expected winnings of a ticket. (b) If a ticket costs $1, what is the expected value of the game?
7.78.A player tosses three fair coins. He wins $5 if three heads occur, $3 if two heads occur, and $1 if only one head occurs.
On the other hand, he loses $15 if three tails occur. Find the value of the game to the player.MEAN,VARIANCE, AND STANDARD DEVIATION
7.79. Find the meanμ, varianceσ^2 , and standard deviationσof each distribution:
(a) x 238(b)y − 10123
f(x)^1 / 4 1 / 2 1 / 4 g(y) 0.3 0.1 0.1 0.3 0.2
7.80. Find the meanμ, varianceσ^2 , and standard deviationσof the following two-point distribution wherep+q=1:x ab
f(x) pq
7.81. LetW=XYwhereXandYare the random variables in Problem 7.33. (RecallW(s)=(XY )(s)=X(s)Y(s).)
Find: (a) the distributionhofW; (b) findE(W).
DoesE(W)=E(X)E(Y)?
7.82. LetXbe a random variable with the distribution:
x −11 2
f(x) 0.2 0.5 0.3
(a) Find the mean, variance, and standard deviation ofX.
(b) Find the distribution, mean, variance, and standard deviation ofYwhere:
(i)Y=X^4 ; (ii)Y= 3 X.