A fair eight-sided die is rolled six times. A success is getting a 6, 7 and 8. Find the probability of getting three successes.
prob(success) in a single toss = 3/8
prob(3 successes in 6 tries)
= C(6,3) (3/8)^3 (5/8)^3
= 20(27/512)(125/512)
= 67500/262144
= 16875/65536
= appr .257
To find the probability of getting three successes when rolling a fair eight-sided die six times, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's find the total number of possible outcomes. Since we are rolling the eight-sided die six times, each roll can have 8 possible outcomes. Since there are six rolls, the total number of possible outcomes is 8^6.
Now, let's find the number of favorable outcomes. We have three successes, which means we need to count the number of ways to obtain a 6, 7, and 8 exactly three times in six rolls.
To calculate this, we can use the concept of combinations. We want to select three positions out of the six rolls to place the successes, and for each success, we have three options (6, 7, or 8). Therefore, the number of favorable outcomes is given by the combination formula:
C(n, k) = n! / (k!(n-k)!)
In this case, n = 6 (number of rolls) and k = 3 (number of successes). Plugging these values into the formula, we calculate:
C(6, 3) = 6! / (3!(6-3)!) = 20
So, the number of favorable outcomes is 20.
Now, we can find the probability of getting three successes by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 20 / (8^6)
Simplifying further:
Probability = 20 / 262,144
So, the probability of getting three successes when rolling a fair eight-sided die six times is approximately 0.0000762939, or about 0.0076%.