A six-sided die is rolled six times. What is the probability that the die will show an even number exactly two times?
so, how many ways are there to pick the two even throws from 6 throws?
To find the probability of getting an even number exactly two times when rolling a six-sided die six times, we need to calculate the number of favorable outcomes divided by the total number of possible outcomes.
1. Determine the number of ways to get an even number exactly two times:
- Choose two positions out of the six rolls to show an even number, which can be done in C(6, 2) = 6! / (2! * (6-2)!) = 15 ways.
- For the remaining four positions, they can show odd numbers, which can be done in 3^4 = 81 ways (as there are three odd numbers on the die).
- So, the number of ways to get an even number exactly two times is 15 * 81 = 1,215 ways.
2. Determine the total number of possible outcomes:
- Each roll has six possible outcomes (numbers 1 to 6).
- Since we are rolling the die six times, the total number of possible outcomes is 6^6 = 46,656.
3. Calculate the probability:
- The probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
- Therefore, the probability is 1,215 / 46,656 ≈ 0.0260806, or approximately 2.6%.
To find the probability of the given scenario, we need to determine the number of favorable outcomes and the number of possible outcomes.
First, let's consider the favorable outcomes. We need to roll an even number exactly two times. The even numbers on a six-sided die are 2, 4, and 6.
To calculate the number of favorable outcomes:
1. Select two positions or rolls where the die shows an even number. This can be done in "6 choose 2" ways, denoted as C(6, 2).
C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15
2. For the remaining four positions or rolls, we need to roll odd numbers. The odd numbers on a six-sided die are 1, 3, and 5.
So, for each of the remaining four positions, there are 3 possible outcomes (odd numbers).
Therefore, the total number of favorable outcomes is 15 * 3^4 = 15 * 81 = 1215.
Next, let's consider the possible outcomes. Since the die is rolled six times, each roll has six possible outcomes (1, 2, 3, 4, 5, or 6).
Therefore, the total number of possible outcomes is 6^6 = 46656.
Finally, to find the probability, we divide the number of favorable outcomes by the number of possible outcomes:
Probability = Favorable outcomes / Possible outcomes
Probability = 1215 / 46656 ≈ 0.0261 (rounded to four decimal places)
So, the probability that a six-sided die will show an even number exactly two times when rolled six times is approximately 0.0261 or 2.61%.
P(even) = 1/2
s, P(even,even) = 1/2 * 1/2 = 1/4