Chapter 13.3, Problem 23AYU

In Problems 23-26, consider the experiment of tossing a coin twice. The table lists six possible assignments of probabilities for this experiment. Using this table, answer the following questions.

Which of the assignments of probabilities is(are) consistent with the definition of a probability model?

To determine

Which of the assignments of probabilities is (are) consistent with the definition of a probability model?

Answer to Problem 23AYU

A, B, C and F represents a probability model.

Explanation of Solution

Given:

Consider the experiment of tossing a coin twice. The table lists six possible assignments of probabilities for this experiment.

Formula used:

Probability of any event is $0\le P\left(E\right)\le 1$.

${\displaystyle \sum }_{i=1}^{n}P\left(e_i\right)=P\left(e_1\right)+P\left(e_2\right)+\cdot \cdot \cdot +P\left(e_n\right)=1$

Calculation:

Consider,

Assignment A.

Probability of each event is $0\le \frac{1}{4}\le 1$.

Total probability is $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1$.

Assignment B.

Probability of each event is $0\le P\left(E\right)\le 1$.

Total probability is $0+0+0+1=1$.

Assignment C.

Probability of each event is $0\le P\left(E\right)\le 1$.

Total probability is $\frac{3}{16}+\frac{5}{16}+\frac{5}{16}+\frac{3}{16}=1$.

Assignment D.

Probability of one of the event is $-\frac{1}{2}\le 1$.

Assignment E.

Probability of each event is $0\le P\left(E\right)\le 1$.

Total probability is $1/4+1/4+1/4+1/8\ne 1$.

Assignment F.

Probability of each event is $0\le P\left(E\right)\le 1$.

Total probability is $1/9+2/9+2/9+4/9=1$.

Hence, A, B, C and F represents a probability model because the probability of each event have values from 0 to 1 (inclusive) and the sum of the probabilities of all the events sums to 1.