A researcher plans to ask 8 randomly selected people if they went to the soccer tournament. The table below shows the probability distribution. Calculate the probability that at most five people went to the tournament.
Positive response
0
1
2
3
4
5
6
7
8
Probability
0.005
0.045
0.04
A
0.3
0.02
0.04
0.2
0.3
To calculate the probability that at most five people went to the tournament, we need to sum up the probabilities of all the responses from 0 to 5.
Probability of 0 people: 0.005
Probability of 1 person: 0.045
Probability of 2 people: 0.04
Probability of 3 people: A (unknown)
Probability of 4 people: 0.3
Probability of 5 people: 0.02
To find the value of A, we need to subtract the probabilities of all the previous responses from 1. A = 1 - (0.005 + 0.045 + 0.04 + 0.3 + 0.02) = 1 - 0.41 = 0.59
Now we can calculate the probability that at most five people went to the tournament:
Probability (at most 5 people) = Probability (0 people) + Probability (1 person) + Probability (2 people) + Probability (3 people) + Probability (4 people) + Probability (5 people)
= 0.005 + 0.045 + 0.04 + 0.59 + 0.3 + 0.02
= 1
Therefore, the probability that at most five people went to the tournament is 1 (or 100%).
To calculate the probability that at most five people went to the tournament, we need to sum the probabilities of the outcomes where there are 0, 1, 2, 3, 4, or 5 people who went.
Probability of 0 people going: 0.005
Probability of 1 person going: 0.045
Probability of 2 people going: 0.04
Probability of 3 people going: A (unknown)
Probability of 4 people going: 0.3
Probability of 5 people going: 0.02
Now we need to solve for the unknown probability, represented as 'A'. Since all probabilities must add up to 1, we can use the equation:
0.005 + 0.045 + 0.04 + A + 0.3 + 0.02 = 1
Simplifying the equation, we have:
0.41 + A = 1
Subtracting 0.41 from both sides, we get:
A = 1 - 0.41 = 0.59
Therefore, the probability of 3 people going to the tournament is 0.59.
Now, we can calculate the probability of at most five people going by summing the probabilities:
Probability of at most 5 people going = 0.005 + 0.045 + 0.04 + 0.59 + 0.3 + 0.02 = 1
So the probability that at most five people went to the tournament is 1.
To calculate the probability that at most five people went to the tournament, we need to sum the probabilities of the outcomes where the number of people who went to the tournament is less than or equal to five.
Looking at the given probability distribution table, we can see that the probabilities for the outcomes where 0, 1, 2, 3, 4, or 5 people went to the tournament are as follows:
0 people: Probability = 0.005
1 person: Probability = 0.045
2 people: Probability = 0.04
3 people: Probability = 0.3
4 people: Probability = 0.02
5 people: Probability = 0.04
To find the probability that at most five people went to the tournament, we simply add up these probabilities:
Probability(at most 5 people) = 0.005 + 0.045 + 0.04 + 0.3 + 0.02 + 0.04
Calculating this sum: 0.005 + 0.045 + 0.04 + 0.3 + 0.02 + 0.04 = 0.45
Therefore, the probability that at most five people went to the tournament is 0.45.