suppose that you roll a pair of 8-sided dice total of 51 times. what is the probability that you will get a sum of 5 at least 3 times
To find the probability of getting a sum of 5 at least 3 times when rolling a pair of 8-sided dice 51 times, we need to consider the total number of ways this event can occur and divide it by the total number of possible outcomes.
Let's break this down step by step:
Step 1: Find the total number of possible outcomes.
Since each 8-sided die has 8 possible outcomes (numbers 1 to 8), the total number of possible outcomes for the pair of dice is 8 * 8 = 64.
Step 2: Find the number of ways to get a sum of 5 at least 3 times.
To calculate this, we can use the concept of combinations. We need to consider all possible scenarios where we get a sum of 5 at least 3 times.
Let's analyze it case by case:
- Case 1: Three instances of sum 5 and the rest any other sum
We can choose 3 out of 51 rolls to get a sum of 5. The remaining 48 rolls can have any other sum, excluding 5. So, we have:
C(51, 3) * (63^(48)) combinations.
- Case 2: Four instances of sum 5 and the rest any other sum
We can choose 4 out of 51 rolls to get a sum of 5. The remaining 47 rolls can have any other sum, excluding 5. So, we have:
C(51, 4) * (63^(47)) combinations.
- Case 3: Five instances of sum 5 and the rest any other sum
We can choose 5 out of 51 rolls to get a sum of 5. The remaining 46 rolls can have any other sum, excluding 5. So, we have:
C(51, 5) * (63^(46)) combinations.
Step 3: Calculate the probability.
The probability is calculated by dividing the number of favorable outcomes (the number of ways to get a sum of 5 at least 3 times) by the total number of possible outcomes.
P = (Number of favorable outcomes) / (Total number of possible outcomes)
P = (C(51, 3) * (63^(48)) + C(51, 4) * (63^(47)) + C(51, 5) * (63^(46))) / (64^51)
By substituting the values in the above expression, you can calculate the probability.