Question

A typical "Pick Six" lottery run by some states and countries works as follows: Six numbered balls are selected (unordered, without replacement) from 49. A player selects at random six balls. Calculate the following probabilities:

a. The player matches exactly three of the six selected balls.

b. The player matches exactly four of the six selected balls.

c. The player matches exactly five of the six selected balls.

d. The player matches exactly six of the six selected balls.

Answer 1

There are 6 correct numbers and 43 incorrect numbers.

P(Event) = Number of favourable outcomes/Total Number of outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

Total Outcomes = 49C3

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(a) Favorable outcomes, choosing 3 correct balls from 6 and 3 incorrect balls from 43 = 6C3 * 43C3

Therefore the required probability = (6C3 * 43C3) / 49C3 = 246820 / 13983816 = 61705 / 3495954 = 0.01765

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(b) Favorable outcomes, choosing 4 correct balls from 6 and 2 incorrect balls from 43 = 6C4 * 43C2

Therefore the required probability = (6C4 * 43C2) / 49C3 = 13545 / 13983816 = 4515 / 4661272 = 0.00097

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(c) Favorable outcomes, choosing 5 correct balls from 6 and 1 incorrect balls from 43 = 6C5 * 43C1

Therefore the required probability = (6C3 * 43C3) / 49C3 = 258 / 13983816 = 43 / 2330636 = 0.000018

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(d) Favorable outcomes, choosing 6 correct balls from 6 and 0 incorrect balls from 43 = 6C6 * 43C0

Therefore the required probability = (6C6 * 43C0) / 49C3 = 1 / 13983816 = 0.000000072

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