SOLUTION: A coin is tossed three times. Let Z denote the number of heads on the first toss and W the total number of heads on the 3 tosses. If the coin is unbalanced and a head has a 40% cha
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Question 1178070: A coin is tossed three times. Let Z denote the number of heads on the first toss and W the total number of heads on the 3 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find the joint probability distribution of W and Z.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's find the joint probability distribution of W and Z step by step.
**Understanding the Variables**
* **Z:** Number of heads on the first toss. Z can be 0 or 1.
* **W:** Total number of heads on the three tosses. W can be 0, 1, 2, or 3.
* **Probability of Heads (H):** P(H) = 0.4
* **Probability of Tails (T):** P(T) = 1 - 0.4 = 0.6
**Possible Outcomes of Three Tosses**
The possible outcomes are:
* HHH
* HHT
* HTH
* HTT
* THH
* THT
* TTH
* TTT
**Calculating Probabilities of Each Outcome**
* HHH: (0.4)(0.4)(0.4) = 0.064
* HHT: (0.4)(0.4)(0.6) = 0.096
* HTH: (0.4)(0.6)(0.4) = 0.096
* HTT: (0.4)(0.6)(0.6) = 0.144
* THH: (0.6)(0.4)(0.4) = 0.096
* THT: (0.6)(0.4)(0.6) = 0.144
* TTH: (0.6)(0.6)(0.4) = 0.144
* TTT: (0.6)(0.6)(0.6) = 0.216
**Creating the Joint Probability Distribution Table**
We need to find P(W = w, Z = z) for all possible values of w and z.
| Z\W | 0 | 1 | 2 | 3 |
|-----|-------|-------|-------|-------|
| 0 | TTT | TTH,THT | THH | N/A |
| 1 | N/A | HTT | HTH,HHT | HHH |
Let's calculate the probabilities:
* **P(W = 0, Z = 0):** TTT = 0.216
* **P(W = 1, Z = 0):** TTH + THT = 0.144 + 0.144 = 0.288
* **P(W = 2, Z = 0):** THH = 0.096
* **P(W = 3, Z = 0):** 0
* **P(W = 0, Z = 1):** 0
* **P(W = 1, Z = 1):** HTT = 0.144
* **P(W = 2, Z = 1):** HTH + HHT = 0.096 + 0.096 = 0.192
* **P(W = 3, Z = 1):** HHH = 0.064
**Joint Probability Distribution Table**
| Z\W | 0 | 1 | 2 | 3 |
|-----|--------|--------|--------|--------|
| 0 | 0.216 | 0.288 | 0.096 | 0 |
| 1 | 0 | 0.144 | 0.192 | 0.064 |
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