Final answer:
The probability distribution of the even number of upper faces when rolling a tetrahedral die 8 times can be calculated using the binomial probability formula. The probability of getting an even number of upper faces on a single roll of the die is 1/2. Using this formula, the probability distribution is: X=0: 1/256, X=2: 7/256, X=4: 35/256, X=6: 7/256, X=8: 1/256.
Step-by-step explanation:
The probability distribution of the even number of upper faces when rolling a tetrahedral die 8 times can be calculated using the binomial probability formula.
The probability of getting an even number of upper faces on a single roll of the die is 2/4 = 1/2, since there are 2 even numbers (2 and 4) out of 4 possible outcomes (1, 2, 3, and 4).
The formula for the probability distribution is P(X=k) = (nCk) * p^k * (1-p)^(n-k), where n is the number of rolls, k is the number of even upper faces, p is the probability of getting an even upper face, and nCk represents the number of combinations of n things taken k at a time.
Using this formula, the probability distribution of the even number of upper faces for 8 rolls of the tetrahedral die is:
P(X=0) = (8C0) * (1/2)^0 * (1-(1/2))^(8-0) = 1 * 1 * (1/2)^8 = 1/2^8 = 1/256
P(X=2) = (8C2) * (1/2)^2 * (1-(1/2))^(8-2) = 28 * 1/4 * 1/2^6 = 7/256
P(X=4) = (8C4) * (1/2)^4 * (1-(1/2))^(8-4) = 70 * 1/16 * 1/2^4 = 35/256
P(X=6) = (8C6) * (1/2)^6 * (1-(1/2))^(8-6) = 28 * 1/64 * 1/2^2 = 7/256
P(X=8) = (8C8) * (1/2)^8 * (1-(1/2))^(8-8) = 1 * 1/256 * 1 = 1/256
Therefore, the probability distribution of the even number of upper faces for 8 rolls of the tetrahedral die is:
X=0: 1/256
X=2: 7/256
X=4: 35/256
X=6: 7/256
X=8: 1/256