Question

In your pocket, you have seven coins: 2 ten cent pieces, 2 fifty cent pieces, 2 one dollar pieces and 1 two dollar piece. You simultaneously choose two coins from your pocket (before you can feel the difference between any of them). Let ? represent the value of the two coins (in cents) that you select from the pocket. a) NEATLY Create a probability distribution for ? and ?(? = ?). HINT: Consider using a tree diagram to represent the value of the two coins before creating the distribution table of the value in cents.

Answer 1

Let us treat this as 2 steps, even though we simultaneously choose 2 coins. The following is the decison tree.

Step 1: Choose the first coin from 7 coins

Node 1:

• Probability of choosing a ten cent is p=2/7
• Probability of choosing a fifty cent is p=2/7
• Probability of choosing a $1 is p=2/7
• Probability of choosing a $2 is p=1/7

step 2: choose the second coin from the remaining 6 coins: The probability and the coins available depends on the choice in step 1

Node 2: Get a 10 cent in the fist step (X is the total value of 2 coins)

• Probability of choosing a ten cent is p=1/6 (only 1 10 cent out is left of 6 remainaing coins). The value of X=10+10=20 cents. The joint probability is Prob(10)*Prob(10)= 2/7*1/6
• Probability of choosing a fifty cent is p=2/6. The value of X=10+50=60 cents. The joint probability is Prob(10)*Prob(50)= 2/7*2/6
• Probability of choosing a $1 is p=2/6. The value of X=10+100=110 cents. The joint probability is 2/7*2/6
• Probability of choosing a $2 is p=1/6. The value of X=10+200=210 cents. The joint probability is 2/7*1/6

Node 3: Get a 50 cent in the fist step (X is the total value of 2 coins)

• Probability of choosing a ten cent is p=2/6 . The value of X=50+10=60 cents. The joint probability is 2/7*2/6
• Probability of choosing a fifty cent is p=1/6(only 1 50 cent is left out of 6 remainaing coins). The value of X=50+50=100 cents. The joint probability is 2/7*1/6.
• Probability of choosing a $1 is p=2/6. The value of X=50+100=150 cents. The joint probability is 2/7*2/6
• Probability of choosing a $2 is p=1/6. The value of X=50+200=250 cents. The joint probability is 2/7*1/6.

Node 4: Get a $1 in the fist step (X is the total value of 2 coins)

• Probability of choosing a ten cent is p=2/6 . The value of X=100+10=110 cents. The joint probability is 2/7*2/6
• Probability of choosing a fifty cent is p=2/6. The value of X=100+50=150 cents. The joint probability is 2/7*2/6
• Probability of choosing a $1 is p=1/6(only 1 $1 is left out of 6 remainaing coins). The value of X=100+100=200 cents. The joint probability is 2/7*1/6.
• Probability of choosing a $2 is p=1/6. The value of X=100+200=300 cents. The joint probability is 2/7*1/6

Node 4: Get a $2 in the fist step (X is the total value of 2 coins)

• Probability of choosing a ten cent is p=2/6 . The value of X=200+10=210 cents. The joint probability is 1/7*2/6
• Probability of choosing a fifty cent is p=2/6. The value of X=200+50=250 cents. The joint probability is 1/7*2/6
• Probability of choosing a $1 is p=2/6. The value of X=200+100=300 cents. The joint probability is 1/7*2/6.
• No $2 coins left to choose in the second step

the following is the list of X and the joint probability of each leaf

X	P(X)
20	0.0476
60	0.0952
110	0.0952
210	0.0476
60	0.0952
100	0.0476
150	0.0952
250	0.0476
100	0.0952
150	0.0952
200	0.0476
300	0.0476
210	0.0476
250	0.0476
300	0.0476

The joint probability would sum to 1

Now we combine the rows corresponding to same values of X.

and get the following probability distribution for X

X	P(X=x)
20	0.0476
60	0.1905
100	0.1429
110	0.0952
150	0.1905
200	0.0476
210	0.0952
250	0.0952
300	0.0952