the participants in a television quiz show are picked from a large pool of applicants with approximately equal number of men and women.Among the last 11 participant there have been only 2 women.If participants are picked randomly,what is the probability of getting 2 or fewer women when 11 people are picked
the answer is 0.033!!
To find the probability of getting 2 or fewer women when 11 people are randomly picked, we can use the binomial distribution formula.
The probability mass function of the binomial distribution is given by:
P(X=r) = C(n,r) * p^r * (1-p)^(n-r)
Where:
P(X=r) is the probability of getting r successes,
C(n,r) is the number of ways to choose r successes out of n trials (given by the combination formula: n! / (r! * (n-r)!)),
p is the probability of success on a single trial (probability of picking a woman), and
(1-p) is the probability of failure on a single trial (probability of picking a man).
In this case, we want to calculate the probability of getting 2 or fewer women, so we need to sum up the probabilities of getting 0, 1, or 2 women.
Let's assume the probability of picking a woman is p, and since there are an equal number of men and women, p = 0.5.
Using the binomial distribution formula, the probability of getting 0 women is:
P(X=0) = C(11,0) * 0.5^0 * (1-0.5)^(11-0)
= 1 * 1 * 0.5^11
= 0.00048828125
The probability of getting 1 woman is:
P(X=1) = C(11,1) * 0.5^1 * (1-0.5)^(11-1)
= 11 * 0.5 * 0.5^10
= 0.005859375
The probability of getting 2 women is:
P(X=2) = C(11,2) * 0.5^2 * (1-0.5)^(11-2)
= 55 * 0.5^2 * 0.5^9
= 0.0546875
Therefore, the probability of getting 2 or fewer women when 11 people are randomly picked is the sum of these three probabilities:
P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2)
= 0.00048828125 + 0.005859375 + 0.0546875
= 0.06103515625
The probability of getting 2 or fewer women is approximately 0.061 or 6.1%.
To calculate the probability of getting 2 or fewer women when 11 people are randomly picked from a pool of applicants with an approximately equal number of men and women, we need to use the concept of binomial probability.
First, let's determine the number of ways to select 2 or fewer women out of 11 participants. We can break it down as follows:
1. Selecting 0 women: There are (11 choose 0) ways of selecting 0 women out of 11 participants, which is equal to 1.
2. Selecting 1 woman: There are (11 choose 1) ways of selecting 1 woman out of 11 participants, which is equal to 11.
3. Selecting 2 women: There are (11 choose 2) ways of selecting 2 women out of 11 participants, which is equal to 55.
Now, let's calculate the total number of ways to select any 11 participants out of the large pool. This can be calculated using the combinations formula:
Total number of ways to select 11 participants = (total number of participants choose 11)
= (number of men + number of women choose 11)
= (approximately equal number of men and women choose 11) [given]
= (2 * (approximately equal number of men) choose 11) [since number of men = number of women]
Now, we can substitute the values and calculate the total number of ways:
Total number of ways = (2 * (approximately equal number of men) choose 11)
= (2 * (approximately equal number of men)!) / (11! * (approximately equal number of men - 11)!)
Finally, we can calculate the probability by dividing the total number of ways to select 2 or fewer women by the total number of ways to select any 11 participants:
Probability = (number of ways to select 0 women + number of ways to select 1 woman + number of ways to select 2 women) / total number of ways
Therefore, the probability of getting 2 or fewer women when 11 people are picked randomly can be calculated using the steps provided above.
Find P(0), P(1), and P(2). Add together for your probability.
You can use a binomial probability table to determine each of the probabilities.