6. Working out Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many times did you go to an exercise or fitness center or work out?” Call the response Y for short. Based on a large sample survey, here is a probability model for the answer you will get:

Days	0	1	2	3	4	5	6	7
Probability	0.68	0.05	0.07	0.08	0.05	0.04	0.01	0.02

(a) Show that this is a legitimate probability distribution.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) Describe the event $Y<7$words. Wthat is$P(Y<7)$?
(d) Express the event “worked out at least once” in terms of $Y\ge 1$. What is the probability of this event?

To show that the probability distribution is a legitimate probability distribution.

Determine the sum of probabilities as follows:

Sum of probabilities $=0.68+0.05+0.08+0.05+0.01+0.02$

=1

The probability range from 0 to ,1 and the sum of the probabilities equals 1.
As a result, the probability distribution provided is a legitimate probability distribution.

To make a histogram of the probability distribution and interpret it.

To histogram of the probability distribution as follows:

The distribution is right-skewed, with the highest bar to the left and a tail of bars to the right.

To describe the event $Y<7$and$P(Y<7)$.

Since,$X<7$ denotes that number of days that people workout is less than 7.
Let, determine $P(X<7)$as follows:

$P(X<7)=P(X=1)+P(X=2)+\dots ..+P(X=6)$

$$\begin{gathered} =0.68+0.05+\dots ..+0.01 \\ =0.98 \end{gathered}$$

As a result, the probability is 0.98

To determine the probability of the event. And to express the event “worked out at least once” in terms of Y.

In terms of Y, the event "exercise at least once" may be expressed as $Y\ge 1$

Then $P(Y\ge 1)$

$P(Y\ge 1)=1-P(Y<0)$

$=1-0.68$

$=0.32$

As a result, the probability is 0.32.