There are 12 E's among the 100 tiles in Scrabble. What is the probability of selecting all 4 E's when selecting 4 tiles?
One answer key says 1/735 while another says 3/23765, but I think it is the second one.
To calculate the probability of selecting all 4 E's when selecting 4 tiles, we need to consider the number of ways this can happen and the total number of possible outcomes.
The number of ways to select all 4 E's can be determined using combinations. Since there are 12 E's in total, the number of ways to select 4 E's is denoted as C(12, 4).
C(n, r) represents the number of ways to select r items from a set of n items, and it is calculated using the formula:
C(n, r) = n! / (r!(n-r)!)
Therefore, in this case:
C(12, 4) = 12! / (4!(12-4)!) = 12! / (4!8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495.
Now, let's calculate the total number of possible outcomes when selecting 4 tiles out of 100. This can be calculated as C(100, 4):
C(100, 4) = 100! / (4!(100-4)!) = 100! / (4!96!) = (100 * 99 * 98 * 97) / (4 * 3 * 2 * 1) = 3,921,225.
Finally, the probability of selecting all 4 E's out of 4 tiles can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes = 495 / 3,921,225 ≈ 0.000126244.
Therefore, the correct probability is approximately 0.000126244, which can be rounded to 3/23765. Hence, the second answer is correct.
To calculate the probability of selecting all 4 E's when selecting 4 tiles, we need to consider the total number of possible outcomes and the number of successful outcomes.
First, let's find the total number of possible outcomes. Since we are selecting 4 tiles from a total of 100, we can use the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, n is the total number of tiles (100) and r is the number of tiles we are selecting (4).
C(100, 4) = 100! / (4!(100-4)!) = 3921225
So there are 3,921,225 possible ways to select 4 tiles from the 100 available.
Next, let's determine the number of successful outcomes, which is the number of ways to select all 4 E's. Since there are 12 E's in the total 100 tiles, we can use the combination formula again:
C(n, r) = n! / (r!(n-r)!)
In this case, n is the total number of E's (12) and r is the number of E's we want to select (4).
C(12, 4) = 12! / (4!(12-4)!) = 495
So there are 495 different ways to select all 4 E's from the available 12.
Now, to calculate the probability, we divide the number of successful outcomes by the total number of possible outcomes:
P(selecting all 4 E's) = successful outcomes / total outcomes
= 495 / 3,921,225
≈ 0.000126138828
So the correct answer is approximately 0.000126138828, which is equivalent to 3/23765.
Therefore, the second answer key you mentioned (3/23765) is correct.