Chapter 16, Problem 1E

Using set notation, write out the sample space for each of the following random experiments.

a. A coin is tossed three times in a row. The observation is how the coin lands (H or T ) on each toss.

b. A basketball player shoots three consecutive free throws. The observation is the result of each free throw ( $s$ for success, $f$ for failure).

c. A coin is tossed three times in a row. The observation is the number of times the coin comes up tails.

d. A basketball player shoots three consecutive free throws. The observation is the number of successes.

To determine

(a)

To calculate:

The Sample Space for a coin tossed three times.

Answer to Problem 1E

Solution:

The sample space is $S=\left\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\right\}$ for a coin tossed 3 times.

Explanation of Solution

Procedure used:

When a coin is tossed three times, the outcome in each toss is either head $\left(H\right)$ or tail $\left(T\right)$.

There is one way to get 3 heads: $\left\{HHH\right\}$

There are 3 ways to get 2 heads: $\left\{HHT,THH,HTH\right\}$

There is 1 way to get 3 tails: $\left\{TTT\right\}$

There are 3 ways to get 1 head:$\left\{HTT,THT,TTH\right\}$

The Sample Space of a sequence of three fair coin tosses is all $2^3$ possible sequences of outcomes:

$S=\left\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\right\}$

So there are 8 events in the sample space. If the coin tossed is fair than each event is equally possible.

Given:

A coin is given for tossing.

Conclusion:

Thus, the sample space is $S=\left\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\right\}$ for a coin tossed 3 times.

To determine

(b)

To calculate:

The Sample Space for three throws of basketball.

Answer to Problem 1E

Solution:

The Sample Space of a sequence of three consecutive free throws is $S=\left\{sss,ssf,sfs,sff,fsf,fss,ffs,fff\right\}$.

Explanation of Solution

Given:

A basketball player shoots three consecutive free throws.

Calculation:

The outcome in each throw is either success $\left(s\right)$ or failure $\left(f\right)$.

There is one way to get 3 successes: $\left\{sss\right\}$

There are 3 ways to get 2 successes: $\left\{ssf,fss,sfs\right\}$

There is 1 way to get 3 failures: $\left\{fff\right\}$

There are 3 ways to get 1 success: $\left\{sff,fsf,ffs\right\}$

The Sample Space of a sequence of three consecutive free throws is all $2^3$ possible sequences of outcomes:

$S=\left\{sss,ssf,sfs,sff,fsf,fss,ffs,fff\right\}$

So there are 8 events in the sample space.

Conclusion:

Thus, the Sample Space of a sequence of three consecutive free throws is $S=\left\{sss,ssf,sfs,sff,fsf,fss,ffs,fff\right\}$.

To determine

(c)

To calculate:

The Sample Space for number of times the coin comes up a tail.

Answer to Problem 1E

Solution:

The Sample Space of the number of times the coin comes up tails is,

$S=\left\{0,1,2,3\right\}$

Explanation of Solution

Given:

A coin is given for tossing.

Calculation:

When a coin is tossed three times, the outcome in each toss is either head $\left(H\right)$ or tail $\left(T\right)$.

Numbers of ways to get tail are:

No tail, one tail, two tails, and three tails in a sequence of three fair coin.

The Sample Space of the number of times the coin comes up tails is,

$S=\left\{0,1,2,3\right\}$

Conclusion:

Thus, the Sample Space of the number of times the coin comes up tails is,

$S=\left\{0,1,2,3\right\}$

To determine

(d)

To calculate:

The Sample Space of the number of times the free throw results in success.

Answer to Problem 1E

Solution:

The Sample Space of the number of times the free throw results in success is,

$S=\left\{0,1,2,3\right\}$

Explanation of Solution

Given:

A coin is given for tossing.

Calculation:

When a ball is thrown three times, the outcome in each throw is either success $\left(s\right)$ or failure $\left(f\right)$.

Numbers of ways to get success:

No free throw results in success, one success, two successes, and three successes in a sequence of three free throws.

The Sample Space of the number of times the free throw results in success is,

$S=\left\{0,1,2,3\right\}$

Conclusion:

Thus, the Sample Space of the number of times the free throw results in success is,

$S=\left\{0,1,2,3\right\}$