Assume that 3 dice are thrown simultaneously. What is the probability that exactly one 4 will come up?
Probability of a 4 is 1/6, while a probability of a non-4 = 5/6.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
1/6 * 5/6 * 5/6 = ?
To find the probability of exactly one 4 coming up when three dice are thrown simultaneously, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
Let's break it down step-by-step:
Step 1: Determine the total number of possible outcomes:
Each die has six sides, so the total number of possible outcomes for one die is 6. Since three dice are thrown simultaneously, the total number of possible outcomes will be 6^3 = 216.
Step 2: Determine the number of favorable outcomes:
We want to find the probability of exactly one 4 coming up. There are three possibilities for where the 4 can appear: on the first die, the second die, or the third die.
For the first die to show a 4 and the other two dice to show any number but 4, we have:
(4, anything but 4, anything but 4) = 1 x 5 x 5 = 25 favorable outcomes.
Similarly, for the second die to show a 4 and the other two dice to show any number but 4, we have:
(anything but 4, 4, anything but 4) = 5 x 1 x 5 = 25 favorable outcomes.
And for the third die to show a 4 and the other two dice to show any number but 4, we have:
(anything but 4, anything but 4, 4) = 5 x 5 x 1 = 25 favorable outcomes.
Step 3: Sum up the favorable outcomes:
Adding up the favorable outcomes from each case, we have a total of 25 + 25 + 25 = 75 favorable outcomes.
Step 4: Calculate the probability:
The probability of exactly one 4 coming up is the number of favorable outcomes divided by the total number of possible outcomes:
P(exactly one 4) = 75 / 216 ≈ 0.3472
Therefore, the probability that exactly one 4 will come up when three dice are thrown simultaneously is approximately 0.3472, or 34.72%.
To solve this problem, we need to calculate the total number of possible outcomes and the number of favorable outcomes, i.e., the outcomes in which exactly one 4 appears.
Step 1: Calculate the total number of possible outcomes.
When three dice are thrown simultaneously, each die has 6 possible outcomes (numbers 1 to 6). Since there are three dice, the total number of possible outcomes is 6 × 6 × 6 = 216.
Step 2: Calculate the number of favorable outcomes.
To find the number of favorable outcomes, we need to consider each die separately. We are only interested in the cases where exactly one die shows a 4.
- Case 1: Die 1 shows a 4, and the other two dice don't show a 4.
The possible outcomes for die 1 = 1 (4)
The possible outcomes for die 2 = 5 (1, 2, 3, 5, 6 without 4)
The possible outcomes for die 3 = 5 (1, 2, 3, 5, 6 without 4)
Total number of outcomes for case 1 = 1 × 5 × 5 = 25
- Case 2: Die 2 shows a 4, and the other two dice don't show a 4.
The possible outcomes for die 1 = 5 (1, 2, 3, 5, 6 without 4)
The possible outcomes for die 2 = 1 (4)
The possible outcomes for die 3 = 5 (1, 2, 3, 5, 6 without 4)
Total number of outcomes for case 2 = 5 × 1 × 5 = 25
- Case 3: Die 3 shows a 4, and the other two dice don't show a 4.
The possible outcomes for die 1 = 5 (1, 2, 3, 5, 6 without 4)
The possible outcomes for die 2 = 5 (1, 2, 3, 5, 6 without 4)
The possible outcomes for die 3 = 1 (4)
Total number of outcomes for case 3 = 5 × 5 × 1 = 25
Step 3: Calculate the probability.
The probability of an event is given by the formula:
Probability = Number of favorable outcomes / Total number of possible outcomes
In this case, the number of favorable outcomes = 25 + 25 + 25 = 75
The total number of possible outcomes = 216
Therefore, the probability that exactly one 4 will come up when three dice are thrown simultaneously is:
Probability = 75 / 216 ≈ 0.3472 (rounded to 4 decimal places)