four people each roll a four die once. find the probability that at least two people will roll the same number?
four people each roll a four die once. find the probability that at least two people will roll the same number?
Answer: To find the probability that at least two people will roll the same number when rolling a four-sided die once, we can use the complementary probability approach. That is, we can find the probability that no two people roll the same number, and subtract this from 1 to get the probability that at least two people will roll the same number.
Let’s first consider the probability that no two people roll the same number. The first person can roll any number, so there are 4 possible outcomes for the first roll. The second person can then roll any number except the number rolled by the first person, so there are 3 possible outcomes for the second roll. Similarly, the third person can roll any number except the numbers rolled by the first two people, so there are 2 possible outcomes for the third roll. Finally, the fourth person can only roll the remaining number, so there is only 1 possible outcome for the fourth roll. Therefore, the probability that no two people roll the same number is:
P(no two people roll the same number) = (4/4) * (3/4) * (2/4) * (1/4) = 3/32
Now, to find the probability that at least two people will roll the same number, we can subtract the probability that no two people roll the same number from 1:
P(at least two people roll the same number) = 1 - P(no two people roll the same number) = 1 - 3/32 = 29/32
Therefore, the probability that at least two people will roll the same number is 29/32 or approximately 0.91.