A fair 6-sided dice is thrown four times and the value Y, the number of times it falls with a 6 showing, is noted.
Show the probability distribution of Y
To find the probability distribution of the random variable Y (number of times a 6 shows up when a fair 6-sided dice is thrown 4 times), we need to calculate the probability for each possible value of Y.
Let's break down the problem step by step:
1. Determine the sample space:
When a 6-sided dice is thrown four times, the sample space consists of all possible outcomes, which is given by 6^4 = 1296. Each outcome is represented by a sequence of four numbers, each ranging from 1 to 6.
2. Define the random variable:
In this case, we define the random variable Y to represent the number of times a 6 shows up in the four throws.
3. Calculate the probability of each outcome:
To find the probability of each outcome, we need to determine the number of favorable outcomes for each possible value of Y.
When Y = 0: The favorable outcome is when no 6 shows up in the four throws. There are 5 choices for each of the four throws (as we need to avoid getting a 6), so the number of favorable outcomes is 5^4 = 625.
When Y = 1: The favorable outcome is when exactly one 6 shows up in the four throws. There are 4 possible positions for the 6 (one in each throw), and for each position, we have 5 choices for the remaining non-6 numbers. So the number of favorable outcomes is 4 * 5^3 = 500.
When Y = 2: The favorable outcome is when exactly two 6s show up in the four throws. There are 6 possible positions for the two 6s (C(4,2) = 6) and for each position, we have 5 choices for the remaining non-6 numbers. So the number of favorable outcomes is 6 * 5^2 = 150.
When Y = 3: The favorable outcome is when exactly three 6s show up in the four throws. As there are only four throws, it is not possible to have three 6s. So the number of favorable outcomes is 0.
When Y = 4: The favorable outcome is when all four throws result in a 6. There is only one such outcome.
4. Calculate the probabilities:
To find the probability for each value of Y, we divide the number of favorable outcomes by the total number of possible outcomes.
P(Y = 0) = 625 / 1296
P(Y = 1) = 500 / 1296
P(Y = 2) = 150 / 1296
P(Y = 3) = 0 / 1296
P(Y = 4) = 1 / 1296
So, the probability distribution of Y is:
Y | P(Y)
---------------
0 | 625 / 1296
1 | 500 / 1296
2 | 150 / 1296
3 | 0 / 1296
4 | 1 / 1296
Note: The sum of the probabilities should always be equal to 1.