From the statement, we know that to win the lottery we must choose 5 numbers from the numbers 1 through 172.
a) Assuming that order is unimportant.
To win, we must pick correctly 5 numbers from 1,...,172.
0. For the first number, we have 5 possibilities from 172, so the probability is 5/172,
,
1. for the second number, we have 4 possibilities from 171, so the probability is 4/171,
,
2. for the third number, we have 3 possibilities from 170, so the probability is 3/170,
,
3. for the fourth number, we have 2 possibilities from 169, so the probability is 2/169,
,
4. for the fifth number, we have 1 possibility from 168, so the probability is 1/168,
To figure out the odds of winning, we multiply together all of the fractional odds of picking each number. Multiplying all the probabilities we get:
So the probability is 1 in 1183009464.
b) Assuming that the order matters.
To win, you have to pick the first number right AND the second number right AND the third number right, etc. In the language of statistics, AND usually means to multiply. So, to figure out the odds of winning, we multiply together all of the fractional odds of picking a given number correctly. The probability of selecting correctly:
• the first number is 1/172,
,
• the second number is 1/171,
,
• the third number is 1/170,
• the fourth number is 1/169,
,
• the fifth number is 1/168,
Multiplying all the probabilities we get:
So the probability is 1 in 141961135680.
Answer
1) The probability of winning the lottery by choosing 5 numbers from 1,...,172 if the order is unimportant, is 1 in 1183009464. or:
2) The probability of winning the lottery by choosing 5 numbers from 1,...,172 if the order matters, is 1 in 141961135680, or:
