7. The hypergeometric distribution Aa Aa In the Hong Kong Lotto, there are 47 ba

Question

7. The hypergeometric distribution Aa Aa In the Hong Kong Lotto, there are 47 balls numbered 1 to 47 in a barrel. To enter the lottery, you select six numbers. Then six balls are randomly drawn from the barrel without replacement to determine the six winning numbers. Select the appropriate distribution in the tool below to help answer some of the following questions Select a Distribution Distributions 0 1 23 The probability that you match exactly one of the winning numbers is the bars in the tool to see exact probabilities.) (Hint: Hover your cursor over To win the jackpot, you must match all six winning numbers. However, most lotteries award small prizes foir matching a subset of the winning numbers. Suppose that you are awarded prize money if you match at least three of the winning numbers. The probability that you will win some prize money is Let x be the number of matches between your six numbers and the winning six numbers. How many different values can the random variable x take on? O 42 O 36 O An infinite number of values The expected value of the random variable x is The standard deviation of the distribution is

Explanation / Answer

There are 6 winning numbers here so matching exactly with one of these 6 numbers is

P(1 of the 6) = 6C1 * 41C5/47C6 = 0.4186

Pr(at least three of the 6 numbers) = Pr(exactly 3 out of 6 match) + Pr(4 out of 6 match) + Pr(5 out of 6 match) + Pr(6 out of 6 match)

= 6C3 * 41C3/47C6 + 6C4 * 41C2/ 47C6 + 6C5 * 41C1/47C6 + 6C6 * 41C0/47C6

= 0.01985551111 + 0.0011455102564 + 0.0000229102051 + 0.0000000931309 = 0.021024

Pr(I will some prize money) = 0.021024

Random variables x can take 7 values from 0 to 6.

Expected value of this variable is

E[X] = 0.7660

Var(X) = E[X2] - E[X}2 = 2.1628 - 0.76602 = 1.5761

Standard deviation of X = 1.2554

Now x is amount of marta's earning

E(x) = Pr(all six match) * Winning all six matches + Pr(no all six match) * losing one dollar

E(x) = 0.000000093131 * 50000000 + (1 - 0.000000093131) * (-1) = $ 3.6565

Here E(X) is greater than $1 so we should say that martha should buy the ticket.

x P(x) xP(X) x^2 P(x) 0 0.4188 0 0 1 0.4188 0.4188 0.418753 2 0.1415 0.2829 1.131764 3 0.0199 0.0596 0.536099 4 0.0011 0.0046 0.073313 5 0.0000 0.0001 0.002864 6 0.0000 0.0000 2.01E-05 Sum 1 0.7660 2.1628

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