Random Variables Practice
1. On Valentine’s Day, a certain restaurant offers a special “Lucky Lover’s Special. When the waiter
brings the check, he also brings the four aces from a deck of cards. The couple selects one of these
cards at random. If they select a black ace, they get no discount. If they select the ace of hearts, they
get a $20 discount. If they select the ace of diamonds, they get to select another card. If it’s the ace
of hearts, they get a $10 discount. Otherwise they get no discount. Let x = the amount of the
discount. Give the probability distribution of x. Also find the values of m x and s x .
2. A game is played with one 10-sided die. The game costs $100 to play. On the first roll, if you roll a
2 you receive $250 and the game is over. If you do not roll a 2 on the first roll, then on the second
roll, if you roll a 5 or 10, you get $150 and the game is over. If not, then on the third roll, if you roll
an even number you get $75, otherwise you receive nothing. Let x = the amount you profit in the
game (so, for example, if you win the $250 on the first roll, you profit $150 since it costs $100 to
play). Construct a probability model for x and give the expected value and standard deviation of x.
3. A man buys a racehorse for $20000 and enters it in two races. He plans to sell the horse afterward,
hoping to make a profit. If the horse wins both races, its value will jump to $100000. IF it wins only
one race, its value will be $50000. If it wins neither race, its value will be only $10000. The owner
believes there is a 20% chance the horse will win the first race and a 30% chance the horse will win
the second race. Assume winning either of the two races is independent of winning the other. Let x =
the profit the man will receive upon selling the horse. Construct a probability model for x and give
the expected value and standard deviation of x.
4. In a group of 10 batteries, 3 are dead. You choose 2 of the 10 batteries at random. Let x = the
number of good batteries you get. Construct a probability model for x and give the expected value and
standard deviation of x.
Answers
1.
x
20
10
0
P(x)
0.25
0.0833
0.6667
m x = $5.83
s x = $8.62
2.
x
150
50
-25
-100
P(x)
0.1
0.18
0.36
0.36
m x = -$21
s x = $78
3.
x
80000
30000
-10000
P(x)
0.06
0.38
0.56
m x = $10600
s x = $25877.40
4.
x
2
1
0
P(x)
0.4667
0.4667
0.0667
m x =1.4 batteries
s x = 0.611 batteries