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Rebecca and her dad went ice-fishing this winter. They caught 18 total fish, including 13 yellow perch. If on the last day of the trip, Rebecca randomly selected 15 fish to donate to fishermen who hadn't caught any fish that day, what is the probability that exactly 11 of the chosen fish are yellow perch? Write your answer as a decimal rounded to four decimal places .

Sagot :

Step 1: Concept

Use the binomial distribution to find the probability that exactly 11 of the chosen fish are yellow perch.

[tex]\begin{gathered} ^nC_rp^rq^{n-r} \\ p\text{ = probability that it is a yellow perch} \\ q\text{ = probability that it is not a yellow perch} \end{gathered}[/tex]

Step 2:

Total = 18

Yellow perch = 13

Others = 5

Step 3:

[tex]\begin{gathered} p\text{ = }\frac{13}{18} \\ q\text{ = }\frac{5}{18} \end{gathered}[/tex]

Step 4:

[tex]\begin{gathered} ^{15}C_{11}\text{ (}\frac{13}{18})^{11}(\frac{5}{18})^4 \\ =\text{ 1365 }\times\text{ 0.027885556 }\times\text{ 0.001653817169} \\ =\text{ }0.2266 \end{gathered}[/tex]

Step 5: Final answer

The probability that exactly 11 of the chosen fish are yellow perch = 0.2266

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