A Mega Millions ticket has you choose 5 numbers out of a list of 70 and then choose 1 Megaball Number from a list of 25 numbers. (19) How many possible combinations are there? (20) What is the probability you will win if you purchase 10,000 tickets?
The correct answer and explanation is:
(19) How many possible combinations are there?
In the Mega Millions lottery, a player must choose 5 numbers from a set of 70, and then choose 1 Megaball number from a set of 25.
To calculate the total number of possible combinations, we first calculate the number of ways to choose the 5 numbers out of the 70. This is done using the combination formula: Combination formula: C(n,r)=n!r!(n−r)!\text{Combination formula: } C(n, r) = \frac{n!}{r!(n-r)!}
Where:
- n=70n = 70 (the total number of numbers available to choose from),
- r=5r = 5 (the number of numbers to choose).
C(70,5)=70!5!(70−5)!=70!5!×65!=70×69×68×67×665×4×3×2×1=12,103,014C(70, 5) = \frac{70!}{5!(70-5)!} = \frac{70!}{5! \times 65!} = \frac{70 \times 69 \times 68 \times 67 \times 66}{5 \times 4 \times 3 \times 2 \times 1} = 12,103,014
So, there are 12,103,014 ways to choose 5 numbers out of 70.
Next, we calculate the number of ways to choose the 1 Megaball number from 25 numbers: C(25,1)=25C(25, 1) = 25
Finally, to get the total number of possible combinations, we multiply the two results: Total combinations=12,103,014×25=302,575,350\text{Total combinations} = 12,103,014 \times 25 = 302,575,350
Thus, the total number of possible combinations is 302,575,350.
(20) What is the probability you will win if you purchase 10,000 tickets?
The probability of winning with a single ticket is the reciprocal of the total number of possible combinations: P(win with one ticket)=1302,575,350P(\text{win with one ticket}) = \frac{1}{302,575,350}
When purchasing 10,000 tickets, the probability of winning with one of the tickets is: P(win with 10,000 tickets)=10,000302,575,350≈0.000033P(\text{win with 10,000 tickets}) = \frac{10,000}{302,575,350} \approx 0.000033
This is approximately 0.000033, or about 1 in 30,257.
Therefore, the probability of winning with 10,000 tickets is extremely low, but it is significantly higher than with just one ticket. Even with 10,000 tickets, the odds of winning are still far from guaranteed.