Chapter 6.3, Problem 11P

Sports: Surfing In Hawaii, January is a favorite month for surfing since 60% of the days have a surf of at least 6 feet (Reference: Hawaii Data Book, Robert C. Schmitt). You work day shifts in a Honolulu hospital emergency room. At the beginning of each month you select your days off, and you pick 7 days at random in January to go surfing. Let r be the number of days the surf is at least 6 feet.

(a) Make a histogram of the probability distribution of r.

(b) What is the probability of getting 5 or more days when the surf is at least 6 feet?

(c) What is the probability of getting fewer than 3 days when the surf is at least 6 feet?

(d) What is the expected number of days when the surf will be at least 6 feet?

(e) What is the standard deviation of the r-probability distribution?

(d) Interpretation Can you be fairly confident that the surf will be at least 6 feet high on one of your days off? Explain.

To determine

To graph: The histogram.

Explanation of Solution

Given: 60% of the days in January have a surf height of at least 6 feet and the number of days randomly picked in January is 7.

Graph:

According to the provided details, the number of days the surf is at least 6 feet high, r follows the binomial distribution with the probability of success in a single trial (p) is 0.60 and the number of trials (n) are 7.

Consider, the probability values provided in Table 2 of the appendix for $n=7$ and $p=0.60$.

The probability values for $r=0,1,2,3,4,5,6,7$ are 0.002, 0.017, 0.077, 0.194, 0.290, 0.261, 0.131, 0.028, respectively.

Follow the steps given below to obtain the histogram:

Step 1: Place the values of r- 0, 1, 2, 3, 4, 5, 6, and 7 on the horizontal axis.

Step 2: Place the values of $P\left(r\right)$- 0.002, 0.017, 0.077, 0.194, 0.290, 0.261, 0.131 and 0.028 on the vertical axis.

Step 3: Construct a bar over each of the r values (0, 1, 2, 3, 4, 5, 6, 7) ranging from $r-0.5$ to $r+0.5$. The height of the corresponding bar is $P\left(r\right)$.

The histogram for the binomial distribution for $n=7$ and $p=0.60$ is shown below:

Interpretation: The graph above displays the fact that the binomial distribution having $n=7$ and $p=0.60$ is bell-shaped. So, the distribution is approximately symmetric.

To determine

To find: The probability of getting 5 or more days when the surf is at least 6 feet high.

Answer to Problem 11P

Solution: The probability is 0.420.

Explanation of Solution

Given: The provided values are: $n=7$ and $p=0.60$.

Calculation: The random variable ‘r’ follows the binomial distribution with the parameters, $n=7$ and $p=0.60$.

The probability of getting 5 or more days when the surf is at least 6 feet high can be calculated by:

$P\left(r\ge 5\right)=P\left(r=5\right)+P\left(r=6\right)+P\left(r=7\right)$

Consider, the probability values provided in the Table 2 of the appendix for $n=7$ and $p=0.60$.

The probability values for $r=5,6,7$ are 0.261, 0.131, 0.028, respectively.

Substitute the values in the above formula,

$\begin{array}{c}P\left(r\ge 5\right)=0.261+0.131+0.028\\ =0.420\end{array}$

The required probability is 0.420.

Interpretation: There is a 42% chance of getting 5 or more days when the surf is at least 6 feet high.

To determine

To find: The probability of getting fewer than 3 days when the surf is at least 6 feet high.

Answer to Problem 11P

Solution: The probability is 0.096.

Explanation of Solution

Given: The provided values are: $n=7$ and $p=0.60$.

Calculation: The random variable ‘r’ follows the binomial distribution with the parameters $n=7$ and $p=0.60$.

The probability of getting fewer than 3 days when the surf is at least 6 feet high can be calculated by:

$\begin{array}{c}P\left(r<3\right)=P\left(r\le 2\right)\\ =P\left(r=0\right)+P\left(r=1\right)+P\left(r=2\right)\end{array}$

Consider, the probability values provided in the Table 2 of the appendix for $n=7$ and $p=0.60$.

The probability values for $r=0,1,2$ are 0.002, 0.017, 0.077, respectively.

Substitute the values in the above formula. Thus,

$\begin{array}{c}P\left(r<3\right)=0.002+0.017+0.077\\ =0.096\end{array}$

The required probability is 0.096.

Interpretation: There is a 9.6% chance of getting fewer than 3 days when the surf is at least 6 feet high.

To determine

To find: The expected number of days when the surf will be at least 6 feet high.

Answer to Problem 11P

Solution: The expected value is 4.2.

Explanation of Solution

Given: The provided values are: $n=7$ and $p=0.60$.

Calculation: The random variable ‘r’ follows the binomial distribution with the parameters, $n=7$ and $p=0.60$.

The formula that is used to calculate the expected value of the binomial distribution is:

$\mu =np$

Substitute the provided values in the above formula,

$\begin{array}{c}\mu =7\times 0.60\\ =4.2\end{array}$

The expected value is 4.2.

Interpretation: One can expect 4.2 out of 7 days when the surf will be at least 6 feet high.

To determine

To find: The standard deviation of the r-distribution.

Answer to Problem 11P

Solution: The standard deviation is 1.296.

Explanation of Solution

Given: The provided values are: $n=7$ and $p=0.60$.

Calculation: The random variable ‘r’ follows the binomial distribution with the parameters, $n=7$ and $p=0.60$.

The formula that is used to calculate the standard deviation of the binomial distribution is:

$\begin{array}{c}\sigma =\sqrt{npq}\\ =\sqrt{np\left(1-p\right)}\end{array}$

Substitute the provided values in the above formula,

$\begin{array}{c}\sigma =\sqrt{7\times 0.60\times \left(1-0.60\right)}\\ =1.296\end{array}$

The standard deviation is 1.296.

Interpretation: The standard deviation of the probability distribution of r is 1.296.

To determine

To explain: Whether one can be fairly confident that the surf will be at least 6 feet high on one of your days off.

Answer to Problem 11P

Solution: Yes, one can be fairly confident as the expected number of days the surf will be at least 6 feet high is 4. The probability of getting at least 1 day out of 7 during which the surf will be at least 6 feet high is 0.998.

Explanation of Solution

Given: The provided values are: $n=7$ and $p=0.60$. Consider the expected value from part (d), that is, $\mu =4.2$.

Calculation: The random variable ‘r’ follows the binomial distribution with the parameters, $n=7$ and $p=0.60$.

The probability of getting at least 1 day out of 7 during which the surf will be at least 6 feet high can be calculated by:

$P\left(r\ge 1\right)=1-P\left(r=0\right)$

Consider, the probability values provided in Table 2 of the appendix for $n=7$ and $p=0.60$.

The probability value for $r=0$ is 0.002.

Substitute the values in the above formula. Thus,

$\begin{array}{c}P\left(r\ge 1\right)=1-0.002\\ =0.998\end{array}$

The probability is 0.998.

Interpretation: The expected number of days that the surf will be at least 6 feet high is approximately 4 and there is a 99.8% chance of getting a surf that is at least 6 feet high on one of the days off out of 7 days. So, one can be confident of getting it.