Chapter 6, Problem 6.54CE

The owner of a charter fishing boat has found that 12% of his passengers become seasick during a half- day fishing trip. He has only two beds below deck to accommodate those who become ill. About to embark on a typical half-day trip, he has 6 passengers on board. What is the probability that there will be enough room below deck to accommodate those who become ill?

To determine

To find:

The probability of having enough room to accommodate the ill

Answer to Problem 6.54CE

The probability of having enough room to accommodate the illis 0.02

Explanation of Solution

Given:

12% of passengers become seasickThere are only two beds to accommodate the ill

There are 6 passengers

Formula used:

The formula for a Binomial Distribution is

The probability of exactly x successes in n trials is

  $P(x)=\frac{n!}{x!(n-x)!}{\pi }^x{(1-\pi )}^{n-x}$

Where n= number of trials

x= the number of successes

  $\pi$ = the probability of success in any given trial

  $(1-\pi )$ = the probability of failure in any given trial

Calculation:

In this problem, a binomial distribution is accurate to use because there are two or more consecutive trials (the six people on board the boat) and there are only two possible, independent outcomes(get sick or not). Using the equation above we can calculate the probability that $P(X>2)$ when x= the number of people who get sick.

  $P(X>2)=P(X=3)+P(X=4)+P(X=5)+P(X=6)$

The calculation for $P(X=3)$ is

  $P(X=3)=\frac{6!}{3!(6-3)!}{(0.12)}^3{(0.88)}^{6-3}$

  $=20\times {(0.12)}^3{(0.88)}^3=0.02$

The calculation for $P(X=4)$ is shown below:

  $P(X=4)=15\times {(0.12)}^4{(0.88)}^2\frac{6!}{4!(6-4)!}{(0.12)}^4{(0.88)}^{6-4}$

  $=15\times {(0.12)}^4{(0.88)}^2=0.002$

Because $P(X=4)$ = approximately zero we can assume that $P(X=5)$ and $P(X=6)$ will also be zero.

Adding up these probabilities we can determine the probability that $P(X=2)$.

  $0.02+0.00=0.02$

Because this probability is so low, it is very unlikely that the owner of the boat will run out of beds below deck.

Conclusion:

Thus, the probability of having enough room to accommodate the illis $0.002$