Prime probability

Probability Level 3

Five distinct numbers are randomly chosen from the first ten prime numbers :

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29.$

What is the probability that the median of the five selected number is 17?

If the probability can be express as $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 33.

1 solution

Chew-Seong Cheong
Apr 26, 2016

The number of ways to chose five numbers from 10 distinct numbers $N = \begin{pmatrix} 10 \\ 5 \end{pmatrix} = 252$ .

For $17$ to be the median, two numbers must be chosen from the six numbers less than $17$ and another two from the three larger than $17$ , therefore the number of ways to choose the five numbers with $17$ as median $P = \begin{pmatrix} 6 \\ 2 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = 15 \times 3 = 45$ .

And its probability $\dfrac{P}{N} = \dfrac{45}{252} = \dfrac{5}{28}$ . Therefore, $a+b = 5 + 28 = \boxed{33}$

But... for 17 to be the median 17 has to be selected, doesn't it? It seems like you don't take that into account, why not?

Ραμών Αδάλια - 4 years, 11 months ago

I did consider it. The number of way to choose 17 to be the median or the third number is 1. Therefore it does not affect the number of ways to choose the five numbers.

Chew-Seong Cheong - 4 years, 11 months ago

Prime probability

The answer is 33.

1 solution

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