Question

use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to determine which of the assignments of probabilities are consistent with the definition of a probability model
is/are consistent with the definition of a probability model
(use a comma to separate answers as needed.)

Explanation:

Step1: Recall the properties of a probability model

A probability model must satisfy two conditions:

1. Each probability \(P(E_i)\geq0\)
2. \(\sum_{i}P(E_i) = 1\)

Step2: Check Assignment A

• Each probability \(P(HH)=\frac{1}{4},P(HT)=\frac{1}{4},P(TH)=\frac{1}{4},P(TT)=\frac{1}{4}\)
• \(P(E_i)\geq0\) for all \(i\)
• \(\sum_{i}P(E_i)=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1\)

Step3: Check Assignment B

• \(P(HH) = 0,P(HT)=0,P(TH)=0,P(TT)=1\)
• \(P(E_i)\geq0\) for all \(i\)
• \(\sum_{i}P(E_i)=0 + 0+0 + 1=1\)

Step4: Check Assignment C

• \(P(HH)=\frac{1}{5},P(HT)=\frac{1}{5},P(TH)=\frac{3}{10},P(TT)=\frac{3}{10}\)
• \(P(E_i)\geq0\) for all \(i\)
• \(\sum_{i}P(E_i)=\frac{1}{5}+\frac{1}{5}+\frac{3}{10}+\frac{3}{10}=\frac{2 + 2+3 + 3}{10}=1\)

Step5: Check Assignment D

• \(P(TH)=-\frac{1}{2}<0\), violates \(P(E_i)\geq0\)

Step6: Check Assignment E

• \(\sum_{i}P(E_i)=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{4}{5}

eq1\)

Step7: Check Assignment F

• \(P(HH)=\frac{1}{14},P(HT)=\frac{5}{14},P(TH)=\frac{5}{14},P(TT)=\frac{3}{14}\)
• \(P(E_i)\geq0\) for all \(i\)
• \(\sum_{i}P(E_i)=\frac{1+5 + 5+3}{14}=1\)

Answer:

A, B, C, F