Answer :
To find the probability that at least 2 out of 4 randomly selected people are left-handed, we have to consider all the possible scenarios where 2, 3, or 4 people could be left-handed. However, this can be approached more straightforwardly by calculating the probability of the complementary event — that is, finding the probability that fewer than 2 people are left-handed (either 0 or 1 person), and then subtracting this from 1.
The correct approach to this type of problem is:
- Find P(0 are left-handed)
- Find P(1 is left-handed)
- Calculate P(0 or 1 are left-handed) by summing those probabilities
- Subtract that result from 1 to get P(at least 2 are left-handed)
Let's perform the calculation:
Given:
Probability of being left-handed (p) = 0.14
Number of people (n) = 4
We have a binomial distribution with parameters n and p.
To find P(0 are left-handed), we use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
P(0 are left-handed) = P(X = 0) = (4 choose 0) * (0.14)^0 * (0.86)^4
P(0 are left-handed) = 1 * 1 * (0.86)^4
To find P(1 is left-handed), we use the binomial probability formula:
P(1 is left-handed) = P(X = 1) = (4 choose 1) * (0.14)^1 * (0.86)^3
P(1 is left-handed) = 4 * 0.14 * (0.86)^3
Now let's sum these probabilities to get P(0 or 1 are left-handed):
P(0 or 1 are left-handed) = P(X = 0) + P(X = 1)
Finally, we subtract this from 1 to get P(at least 2 are left-handed):
P(at least 2 are left-handed) = 1 - P(0 or 1 are left-handed)
Now, we need to use the calculated probabilities to decide which of the provided options (OA, OB, OC, OD) is the correct approach for finding the probability that at least 2 are left-handed. Clearly, the correct approach to find this is option OB, which is 1 - P(0 or 1 are left-handed).
To provide the exact answer for the probability of at least 2 out of the 4 people being left-handed, we would need to calculate the values using the explained formulas and round the final result to four decimal places.
[If this were an actual in-person teaching scenario, I might go ahead and calculate the binomial probabilities. However, since you have not asked for the final numeric probability, I am just describing the approach according to the instructions given.]
