In a recent study 35 percent of people surveyed indicate chocolate was their favorite flavor of ice cream. Suppose we select a sample of ten people and ask them their favorite flavor.
A: What is the probability that four or more name chocolate?
this is a binary probability ... chocolate (c) or not chocolate (n)
... p(c) = .35 ... p(n) = .65
(n + c)^10 = n^10 + 10 n^9 c + 45 n^8 c^2 + 84 n^7 c^3 + ... + c^10
sum the 1st four terms and subtract from 1
... 1 - [.65^10 + (10 * .65^9 * .35) + (45 * .65^8 * .35^2) + (84 * .65^7 * .35^3)]
To find the probability that four or more people name chocolate as their favorite flavor of ice cream, we need to calculate the probability of having 4, 5, 6, 7, 8, 9, or 10 people out of the sample selecting chocolate.
To do this, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of having exactly k successes (in this case, people naming chocolate as their favorite flavor).
- n is the number of trials (number of people in the sample = 10).
- k is the number of successes (number of people naming chocolate as their favorite flavor).
- p is the probability of success (probability of a randomly selected person naming chocolate as their favorite flavor = 0.35).
- (1 - p) is the probability of failure (probability of a randomly selected person not naming chocolate as their favorite flavor = 0.65).
- C(n, k) is the binomial coefficient, which calculates the number of ways to choose k successes from n trials.
Now, let's calculate the probability for each case (k = 4, 5, 6, 7, 8, 9, and 10) and sum them up:
P(X = 4) = C(10, 4) * 0.35^4 * 0.65^6
P(X = 5) = C(10, 5) * 0.35^5 * 0.65^5
P(X = 6) = C(10, 6) * 0.35^6 * 0.65^4
P(X = 7) = C(10, 7) * 0.35^7 * 0.65^3
P(X = 8) = C(10, 8) * 0.35^8 * 0.65^2
P(X = 9) = C(10, 9) * 0.35^9 * 0.65^1
P(X = 10) = C(10, 10) * 0.35^10 * 0.65^0
Finally, sum up all these probabilities to get the final result:
P(X >= 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
To find the probability that four or more people name chocolate as their favorite flavor, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of trials or observations
- k is the number of successful outcomes
- p is the probability of success on a single trial
- (nCk) is the number of combinations of n things taken k at a time (also known as binomial coefficient)
In this case, n = 10 (sample size) and p = 0.35 (probability of choosing chocolate).
Let's calculate the probability using this formula for k = 4, 5, 6, 7, 8, 9, and 10, and then sum them up:
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Using a calculator or statistical software that can calculate binomial probabilities, we can substitute these values into the formula to get the final result.