According to the IRS, 70% of Americans file their taxes electronically.
For a group of 12 American adults who were selected at random, what is the probability that at least five of them file their taxes electronically?
To find the probability that at least five out of twelve American adults file their taxes electronically, we need to use the binomial probability formula. This formula calculates the probability of a certain number of successes in a fixed number of repeated trials, assuming that each trial is independent and has the same probability of success.
The formula is as follows:
P(X = k) = C(n, k) * 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 (12 in this case),
- k is the number of successes (in this case, at least 5),
- p is the probability of success in one trial (0.70, as stated in the question),
- C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k!(n - k)!).
To find the probability of at least five individuals filing their taxes electronically, we need to sum up the probabilities of having exactly 5, 6, 7, 8, 9, 10, 11, or 12 successes. We can calculate each of these probabilities using the formula mentioned above, and then sum them up to get the final probability.
Let's calculate the probabilities for each case:
P(X = 5) = C(12, 5) * (0.70)^5 * (1 - 0.70)^(12 - 5)
P(X = 6) = C(12, 6) * (0.70)^6 * (1 - 0.70)^(12 - 6)
P(X = 7) = C(12, 7) * (0.70)^7 * (1 - 0.70)^(12 - 7)
P(X = 8) = C(12, 8) * (0.70)^8 * (1 - 0.70)^(12 - 8)
P(X = 9) = C(12, 9) * (0.70)^9 * (1 - 0.70)^(12 - 9)
P(X = 10) = C(12, 10) * (0.70)^10 * (1 - 0.70)^(12 - 10)
P(X = 11) = C(12, 11) * (0.70)^11 * (1 - 0.70)^(12 - 11)
P(X = 12) = C(12, 12) * (0.70)^12 * (1 - 0.70)^(12 - 12)
Finally, we can find the probability of at least five individuals filing their taxes electronically by summing up these probabilities:
P(at least 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
Now, you can substitute the values into the formula and calculate the probability.