Q1
We are to find the probability that when three dice are rolled at the same time, the largest value of the three numbers rolled is 4. Let A be the outcome in which the largest number is 4, let B the outcome in which the largest number is 4 or less, and C be the outcome in which the largest number is 3 or less.
Let P(X) denote the probability that the outcome of an event is X. Then
(1) P(B) = …
(2) P(C) = …
(3) Since B = A ∪ C and the outcomes A and C are mutually exclusive, it follows that P(A) = …
Define the events and total outcomes
Event A: The largest number rolled is exactly 4. This means at least one die shows 4, and no die shows 5 or 6.
Event B: The largest number rolled is 4 or less. This means all three dice show numbers from 1 to 4.
Event C: The largest number rolled is 3 or less. This means all three dice show numbers from 1 to 3.
Each die has 6 equally likely outcomes, and with three dice, every combination is equally probable, so we’ll compute probabilities by counting favorable outcomes and dividing by 6³ = 216.
(1) Compute P(B)
Event B occurs when all three dice show a number that is 4 or less (i.e., 1, 2, 3, or 4). For each die:
- Possible outcomes: [1, 2, 3, 4}
- Number of choices per die: 4
Since the dice are independent:
Number of outcomes for three dice: 4 × 4 × 4 = 4³ = 64
Total outcomes: 216
Thus, the probability is: \(\text{P(B)} = \dfrac{64}{216}\)
(2) Compute P(C)
Event C occurs when all three dice show a number that is 3 or less (i.e., 1, 2, or 3). For each die:
- Possible outcomes: [1, 2, 3}
- Number of choices per die: 3
Since the dice are independent:
Number of outcomes for three dice: 3 × 3 × 3 = 3³ = 27
Total outcomes: 216
Thus, the probability is: \(\text{P(C)} = \dfrac{27}{216}\)
(3) Compute P(A) using the given relationship
B = A ∪ C and the outcomes A and C are mutually exclusive.
In probability, for mutually exclusive events, P(A ∩ C) = 0
\(\text{P(B) = p(A) + P(C)}\)
\(\dfrac{64}{216} = \text{P(A)} + \dfrac{27}{216}\)
\(\text{P(A)} = \dfrac{64}{216}\:-\: \dfrac{27}{216}\)
\(\text{P(A)} =\dfrac{37}{216}\)