Ping pong balls numbered from 1 to 18 are placed in a lottery machine and nine balls are selected at random from the machine. No ball is returned to the machine after it is selected. Before the drawing, a lottery player selects seven numbers that he or she predicts will be among those selected. The player wins if all seven numbers they selected are among the nine that are selected by the machine. What are the odds of winning this lottery
The Correct Answer and Explanation is:
To find the odds of winning this lottery, we need to calculate the probability that the player’s seven selected numbers are among the nine drawn by the lottery machine.
Total Number of Ways to Select 9 Balls:
The total number of ways to select 9 balls from the 18 balls available is given by the combination formula:Total Ways=(189)=18!9!(18−9)!=18!9!9!=48,620\text{Total Ways} = \binom{18}{9} = \frac{18!}{9!(18-9)!} = \frac{18!}{9!9!} = 48,620Total Ways=(918)=9!(18−9)!18!=9!9!18!=48,620
This represents all possible outcomes where the machine selects 9 balls from the 18 balls.
Number of Favorable Outcomes:
The player must have their 7 chosen numbers included in the 9 balls selected by the machine. This means that, out of the 9 balls drawn, 7 must be the player’s selected numbers, and the remaining 2 balls must come from the 11 numbers that the player did not select.
The number of ways to choose the 2 balls from the 11 non-selected numbers is:Favorable Ways=(112)=11!2!(11−2)!=11×102×1=55\text{Favorable Ways} = \binom{11}{2} = \frac{11!}{2!(11-2)!} = \frac{11 \times 10}{2 \times 1} = 55Favorable Ways=(211)=2!(11−2)!11!=2×111×10=55
Thus, there are 55 favorable outcomes where the player’s 7 numbers are all among the 9 selected.
Probability of Winning:
The probability of winning is the ratio of favorable outcomes to the total outcomes:P(win)=5548,620≈0.00113P(\text{win}) = \frac{55}{48,620} \approx 0.00113P(win)=48,62055≈0.00113
Odds of Winning:
The odds of winning are calculated by taking the ratio of the probability of winning to the probability of losing. The probability of losing is:P(lose)=1−P(win)=1−0.00113≈0.99887P(\text{lose}) = 1 – P(\text{win}) = 1 – 0.00113 \approx 0.99887P(lose)=1−P(win)=1−0.00113≈0.99887
Thus, the odds of winning are:Odds of Winning=P(win)P(lose)=0.001130.99887≈0.00113\text{Odds of Winning} = \frac{P(\text{win})}{P(\text{lose})} = \frac{0.00113}{0.99887} \approx 0.00113Odds of Winning=P(lose)P(win)=0.998870.00113≈0.00113
So the odds of winning are about 1 in 885.
