a fair die is tossed three times. Find the probability of getting
a. Three 4s
b. Three 5s
c. Three 4s or three 5s
To find the probability of getting specific outcomes when tossing a fair die, we need to know the total number of possible outcomes and the number of favorable outcomes for each event.
a. Probability of getting three 4s:
There is only one favorable outcome, which is getting a 4, so the probability of getting a 4 on any one toss is 1/6. Since there are three independent tosses, we multiply the probabilities together:
P(Three 4s) = (1/6) * (1/6) * (1/6) = 1/216
Therefore, the probability of getting three 4s is 1/216.
b. Probability of getting three 5s:
Similarly, there is only one favorable outcome, getting a 5, so the probability of getting a 5 on any one toss is also 1/6.
P(Three 5s) = (1/6) * (1/6) * (1/6) = 1/216
Therefore, the probability of getting three 5s is also 1/216.
c. Probability of getting three 4s or three 5s (at least one favorable outcome):
To find the probability of getting three 4s or three 5s, we can add the probabilities of each event:
P(Three 4s or Three 5s) = P(Three 4s) + P(Three 5s)
= 1/216 + 1/216
= 2/216
= 1/108
Therefore, the probability of getting three 4s or three 5s is 1/108.
To find the probability of getting specific outcomes when a fair die is tossed multiple times, we need to consider the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes when a fair die is tossed three times can be calculated by multiplying the number of outcomes for each toss. Since a fair die has 6 faces, each with a different number, the total number of outcomes for each toss is 6.
Therefore, the total number of possible outcomes when the die is tossed three times is 6 x 6 x 6 = 216.
Now, let's calculate the number of favorable outcomes for each scenario:
a. Three 4s: Since we want to get three 4s, there is only one favorable outcome: (4, 4, 4).
b. Three 5s: Similarly, there is only one favorable outcome for getting three 5s: (5, 5, 5).
c. Three 4s or three 5s: To calculate the number of favorable outcomes for this scenario, we need to sum up the number of favorable outcomes for each individual scenario. So, the total number of favorable outcomes is 1 (for three 4s) + 1 (for three 5s) = 2.
Finally, we can find the probability of each scenario by dividing the number of favorable outcomes by the total number of possible outcomes:
a. Probability of getting three 4s = (Number of favorable outcomes) / (Total number of possible outcomes) = 1 / 216.
b. Probability of getting three 5s = (Number of favorable outcomes) / (Total number of possible outcomes) = 1 / 216.
c. Probability of getting three 4s or three 5s = (Number of favorable outcomes) / (Total number of possible outcomes) = 2 / 216.
These are the probabilities for the given scenarios.
(a) 1/6^3
(b) ditto
(c) (a)+(b)