The annual art competition in Tiruvannamalai had 10 talented participants, including 6 boys and 4 girls. For the grand finale, a group of 5 must be selected with at least two girls in the final round. What is the probability that a randomly chosen group of 5 will have at least two girls? Round off your answer to 3 decimal places.
Answer (1)
To solve this problem, we need to determine the probability that a randomly chosen group of 5 participants from a pool of 10 (comprising 6 boys and 4 girls) has at least two girls.
Let's break it down step-by-step:
Total Ways to Select 5 Participants:
The total number of ways to select any 5 participants from the 10 participants can be calculated using the combination formula:
( 5 10 ) = 5 × 4 × 3 × 2 × 1 10 × 9 × 8 × 7 × 6 = 252
Ways with at Least Two Girls:
We'll calculate the number of groups with exactly 2, 3, and 4 girls, as any of these groups will meet the criteria of having at least two girls.
Exactly 2 Girls:
Select 2 girls from 4 and 3 boys from 6:
( 2 4 ) × ( 3 6 ) = 6 × 20 = 120
Exactly 3 Girls:
Select 3 girls from 4 and 2 boys from 6:
( 3 4 ) × ( 2 6 ) = 4 × 15 = 60
Exactly 4 Girls:
Select 4 girls from 4 and 1 boy from 6:
( 4 4 ) × ( 1 6 ) = 1 × 6 = 6
Combine these to find the total number of desirable groups:
120 + 60 + 6 = 186
Probability Calculation:
The probability is the number of favorable outcomes divided by the total number of outcomes:
Probability = 252 186 ≈ 0.738
So, the probability that a randomly chosen group of 5 will have at least two girls is approximately 0.738 or 73.8% .
