Let's begin with a rather straightforward lottery in which players select four digits. The winning number is commonly selected in the following fashion: 10 Ping Pong balls (each with a digit 0 through 9 written on it) are placed in a hopper. One Ping Pong ball is selected. That digit now goes in the first place. That Ping Pong ball is put back in the hopper and the process is repeated three times.
What is the probability of picking a winning number?
Then suppose the Ping Pong ball was not replaced. Now what is the probability?
10 cases for 1st draw, 10 for the 2nd, 10 for the third, 10 for the 4th
prob = 1/10^4 = 1/10000
if ball is not returned
prob = 1/(10*9*8*7) = 1/5040
To calculate the probability of winning in the lottery, we need to consider the number of favorable outcomes (winning numbers) and the total number of possible outcomes.
1. Probability with Replacement:
When the Ping Pong ball is replaced after each selection, the probability of picking a winning number remains constant for each digit picked.
a) For the first digit: There are 10 possible digits that can be selected, out of which only one is the winning digit. Therefore, the probability of picking the winning digit in the first place is 1/10.
b) Similarly, for each subsequent digit, the probability of picking the winning digit remains 1/10.
Since all the digits are picked independently, the overall probability of picking a winning number with replacement can be calculated by multiplying the probabilities for each digit:
P(with replacement) = P(first digit) × P(second digit) × P(third digit) × P(fourth digit)
P(with replacement) = (1/10) × (1/10) × (1/10) × (1/10)
P(with replacement) = 1/10,000
Therefore, the probability of picking a winning number with replacement is 1 in 10,000.
2. Probability without Replacement:
When the Ping Pong ball is not replaced after each selection, the probability of picking a winning number changes for each subsequent digit.
a) For the first digit: The same as before, there is a 1/10 chance of picking the winning digit.
b) For the second digit: After one digit has been picked, there are only 9 remaining digits in the hopper. So, the probability of picking the winning digit in the second place is 1/9.
c) For the third digit: After two digits have been picked, there are only 8 remaining digits in the hopper. The probability of picking the winning digit now becomes 1/8.
d) For the fourth digit: With three digits already picked, only 7 digits remain. Hence, the probability of picking the winning digit in the fourth place is 1/7.
Again, multiplying the probabilities for each digit gives us the overall probability of picking a winning number without replacement:
P(without replacement) = P(first digit) × P(second digit) × P(third digit) × P(fourth digit)
P(without replacement) = (1/10) × (1/9) × (1/8) × (1/7)
P(without replacement) = 1/5040
Therefore, the probability of picking a winning number without replacement is 1 in 5040.
To find the probability of picking a winning number, we need to determine the total number of possible outcomes and the number of favorable outcomes.
In this lottery game, there are 10 possible digits (0-9) for each of the four positions. Since each digit is selected independently and with replacement, the total number of possible outcomes is:
Total number of possible outcomes = 10 x 10 x 10 x 10 = 10,000
Now, to calculate the number of favorable outcomes, we need to consider that there is only one winning number, which consists of four specific digits. Since each digit is selected independently and with replacement, the number of favorable outcomes is:
Number of favorable outcomes = 1
Therefore, the probability of picking a winning number in this lottery game is:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 10,000
So, the probability of picking a winning number is 1 in 10,000.
Now, let's consider the case where the Ping Pong ball is not replaced after each selection. In this case, the probability calculations will be different.
For the first selection, there are still 10 possible digits. However, for the second selection, since the previous digit is not replaced, there are only 9 possible digits remaining (as one digit has already been selected). Similarly, for the third selection, there are 8 possible digits remaining, and for the fourth selection, there are 7 possible digits remaining.
Therefore, the new total number of possible outcomes is:
Total number of possible outcomes = 10 x 9 x 8 x 7 = 5,040
The number of favorable outcomes remains the same (1) since there is still only one winning number.
So, the probability of picking a winning number when the Ping Pong ball is not replaced is:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 5,040
Thus, the probability of picking a winning number when the Ping Pong ball is not replaced is 1 in 5,040.