Question

Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer.

(a) Roll a die and record the count of spots on the up-face: P(1) = 0, P(2) = 1/6, P(3) = 1/3, P(4) = 1/3,

P(5) = 1/6, P(6) = 0.

(b) Choose a college student at random and record gender and enrollment status: P(female full-time) = 0.56, P(male full-time) = 0.44, P(female part-time) = 0.24, P(male part-time) = 0.17.

(c) Deal a card from a shuffled deck: P(clubs) = 12/52, P(diamonds) = 12/52, P(hearts) = 12/52,

P(spades) = 16/52.

Short answer

Part (a) The distribution is legitimate.

Part (b) The distribution is not legitimate.

Part (c) The distribution is legitimate.

Step-by-step solution

Part (a)  Step 1. Given Information 

The reported probabilities are presented below after rolling a $6$ -sided die. $P\left(1\right)=0,P\left(2\right)=1/6,P\left(3\right)=1/3,P\left(4\right)=1/3,P\left(5\right)=1/6,P\left(6\right)=0$

Part (a) Step 2. Concept Used  

An event is a subset of an experiment's total number of outcomes. The ratio of the number of elements in an event to the number of total outcomes is the probability of that occurrence.

Part (a)  Step 3. Calculation   

The probability distribution must meet the following conditions: 1) The probabilities must be$0\le p\le 1$ 2)The sum of all probabilities should equal one. All the probabilities in the given distribution are between $0$ and 1 $0+\frac{1}{6}+\frac{1}{3}+\frac{1}{3}+\frac{1}{6}+0=1$ is the total number of probabilities As a result, both conditions are met.

Part (b) Step 1. Calculation   

The conditions of the probability distribution are: $0\le p\le 1$ 1)The probabilities should be $0\le p\le 1$ 2)Sum of all the probabilities should be one. Forgiven distribution, all the probabilities are between $0$and $1$Sum of the probabilities: $0.56+0.44+0.24+0.17=1.41$ Hence, the second condition does not satisfy.

Part (c) Step 1. Calculation  

The conditions of the probability distribution are: 1)The probabilities should be $0\le p\le 1$ 2)Sum of all the probabilities should be one. Forgiven distribution, all the probabilities are between $0$ and $1$ Sum of the probabilities: $\frac{12}{52}+\frac{12}{52}+\frac{12}{52}+\frac{16}{52}=1$ Hence, the second condition does not satisfy.