Six 6-sided dice are rolled. What is the probability that three of the dice show prime numbers and the rest show composite numbers?
The answer is NOT 5/72
answer is 5/54
To solve this problem, we need to determine the total number of outcomes and the number of favorable outcomes.
Total Number of Outcomes:
Each of the six dice has six possible outcomes since they have six sides and each side has a different number. Therefore, the total number of outcomes for all six dice is 6 x 6 x 6 x 6 x 6 x 6 = 6^6.
Number of Favorable Outcomes:
We need to consider the number of prime numbers and composite numbers on each dice separately.
For a dice to show a prime number, it must show one of the numbers 2, 3, or 5. There are three prime numbers out of six possible numbers, so the number of favorable outcomes for each dice is 3.
For a dice to show a composite number, it must show one of the numbers 1, 4, or 6. There are three composite numbers out of six possible numbers, so the number of favorable outcomes for each dice is 3.
Since we want three dice to show prime numbers and the other three dice to show composite numbers, we can determine the number of favorable outcomes by multiplying the number of favorable outcomes for each dice: 3 x 3 x 3 x 3 x 3 x 3 = 3^6.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Number of Favorable Outcomes / Total Number of Outcomes
= 3^6 / 6^6
= 729 / 46656
≈ 0.015625
Therefore, the probability that three of the dice show prime numbers and the rest show composite numbers is approximately 0.015625 or 1.5625%.
To find the probability of rolling three prime numbers and three composite numbers when rolling six 6-sided dice, we need to determine the total number of possible outcomes and the number of desirable outcomes.
First, let's calculate the total number of possible outcomes. Each of the six dice has 6 possible outcomes, so the total number of outcomes will be 6 raised to the power of 6 (6^6), which equals 46,656.
Next, let's determine the number of desirable outcomes, i.e., the number of ways we can get three prime numbers and three composite numbers.
To get three prime numbers, we need to choose 3 out of the 6 dice. There are 6 choose 3 (6C3) ways to do this, which can be calculated using the combination formula:
6C3 = 6! / (3! * (6-3)!)
= 6! / (3! * 3!)
= (6 * 5 * 4) / (3 * 2 * 1)
= 20
Similarly, to get three composite numbers, we need to choose 3 out of the remaining 3 dice. There are 3 choose 3 (3C3) ways to do this, which is equal to 1.
Since these events are independent, we can multiply the favorable outcomes together:
Number of favorable outcomes = 20 * 1 = 20
Therefore, the probability of rolling three prime numbers and three composite numbers is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 20 / 46,656
≈ 0.000428161
So, the probability of rolling three prime numbers and three composite numbers with six 6-sided dice is approximately 0.000428161, not 5/72.
probability of prime (1, 2, 3, 5) ... 4/6 = 2/3
... not prime, composite (4, 6) ... 2/6 = 1/3
(p + n)^6 = p^6 + 6 p^5 n + 15 p^4 n^2 + 20 p^3 n^3 + ...
probability 3 prime and 3 not ... 20 * (2/3 * 1/3)^3