STAT 113 Week 8 (Feb 25)
Work Sheet 5: Chapter 17, 18 & 20
 Probability model and expected value
1. The probability distribution for a game of tossing a fair coin 4 times and recording
the number of heads is:
outcome
No heads
1 head
2 heads
3 heads
4 heads
probability
1/16
4/16
6/16
4/16
?
(a) What must be the probability of 4 heads?
1-1/16-4/16-6/16-4/16=1/16=0.0625
(b) What's the probability of getting at least 2 heads in this game?
sum up the probabilities: 6/16+4/16+1/16=11/16=0.6875
or subtract from total: 1-1/16-4/16=11/16=0.6875
(c) Write out 3 reasons why this is a legitimate probability model:
1. Every possible outcome is listed.
2. Each individual probability is between 0 and 1.
3. All the probabilities added together total to 1.
(d) What‘s the expected number of heads when we toss a fair coins 4 times?
0(1/16)+1(4/16)+2(6/16)+3(4/16)+4(1/16) = 2
2. A deck of cards contains 52 cards, of which 4 are aces. Also there are four
different suits in a deck of 52 cards, 13 cards for each (spade, club, heart, and
diamond). Randomly draw one card from 52.
(a) What's the probability of drawing an ace?
4/52=0.0769
(b) What's the probability of drawing a spade?
13/52=0.25
(c) What's the probability of drawing a diamond or a club?
26/52=0.5
(d) What's the probability of not drawing a heart?
1-13/52=0.75
(e) You win $100 if the card drawn is a 10, and you lose $10 if the card drawn is not a
10. What are your expected winnings?
Expected Winnings = 100(1/13) – 10(12/13)
= -20/13=-1.5385
3. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are
colored green, 18 of the others are red, and 18 are black. The dealer spins the
wheel and at the same time rolls a small ball along the wheel in the opposite
direction. The wheel is carefully balanced so that the ball is equally likely to land
in any slot when the wheel slows. A “fair game” is one in which the expected
value of winnings is 0.
(a) How much would you have to be offered for a win to make betting on black
worthwhile if you lose $10 for a non-black outcome?
Win($)
x
probability 18/38
10
20/38
Fair game if expected Winnings = 0
=> x(18/38) –10(20/38) = 0=> x=11.11
You have to be offered at least $11.11 for a win to make playing this
game worthwhile.
(b) How much would you have to be offered for a win to make betting on green
worthwhile if you lose $25 for a non-green outcome?
Win($)
x
25
probability
2/38
36/38
Fair game if expected Winnings = 0
=>x(2/38) –25(36/38)= 0 => x=450
You have to be offered at least $450 for a win to make playing this game
worthwhile.
4. Suppose we roll two six-sided dice (one blue, one green) and each side of the die
is equally likely to occur. Once we roll two dice, we record the sum of the spots
on the up-faces of the dice.
(a) List all possible sums of two dice in the table below.
Roll 1 (blue die)
1
Roll 2
(green
die)
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
(b) What is the probability model of the sum of two dice rolls?
sum
prob
2
1
36
3
2
36
4
3
36
5
4
36
6
5
36
7
6
36
8
9
5
36
4
36
10
3
36
11
2
36
(c) What is the expected value of the sum of two dice rolls?
E(Sum)=2*1/36+3*2/36+…+11*2/36+12*1/36=7
12
1
36