Question

Asked Jul 2, 2019

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The state runs a lottery once every week in which six numbers are randomly selected from 17 without replacement. A player chooses six numbers before the state’s sample is selected. The player wins if all 6 numbers match. What is the probability that exactly 3 of the 6 numbers chosen by a player appear in the state’s sample? Report answer to 3 decimal places.

Step 1

**Hyper geometric distribution:**

A hyper geometric distribution is a discrete probability distribution that determines the probability of getting *k *successes in *n *draws (without replacement) from a finite population of size *N *that contains exactly *K* success states.

Denote the total number of successes as *k,*

Denote the total number of objects that are drawn without replacement as *n,*

Denote the population size as *N,*

Denote the total number of success states in the population as *K.*

The probability distribution of *k* is a hyper geometric distribution with parameters (*N, K, n*) and the probability mass function (pmf) of *k* is given as:

Step 2

**Parameters of hyper geometric distribution for the given situation:**

In a state’s lottery, 6 numbers are randomly selected from a total of 17 numbers without replacement.

Here, the population size is *N *=17,

The number of draws is *n* = 6.

Furthermore, it is given that, a player chooses 6 numbers before the state’s sample is selected and the player wins, if all the 6 numbers match with the state’s sample.

Thus, the number of success states in the population is *K *= 6.

The probability distribution of *k* successes is a hyper geometric distribution with parameters (*N = *17*, K = *6*, n* = 6) and the probability mass function (pmf) of *k* is given as:

Step 3

**Probability that exactly 3 of the 6 numbers chosen by a player appear in the state’s sample:**

It is given that, 3 of the 6 chosen numbers have to appear in the state’s lottery.

That is, the number of successes is ...

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