At the annual art competition in Tiruvannamalai, there were 10 talented participants, out of which 6 were boys and 4 were girls. As the event reached its climax, the chief Kavitha announced: "For the grand finale, we must select a group of 5. But remember, our jury would love to see at least two girls in the final round!" As the audience waited in anticipation, Pranav, last year's winner, turned to his friend, Gopal and whispered, "What's the probability that the randomly chosen group of 5 has at least two girls?" If Gopal calculated the correct answer, what answer did he find? Note: Round off your answer to 3 decimal places. For example, if the answer is 0.5 put 0.500, if the answer is 0.5447 then put 0.545 and if the answer is 0.7825 then put 0.783.
Answer (1)
In this problem, we want to calculate the probability that a randomly chosen group of 5 participants from a total of 10 participants (6 boys and 4 girls) contains at least two girls.
To solve this problem, we'll use combinations to find the number of ways to choose the group of participants and then apply probability rules.
Step-by-step solution:
Calculate the total number of ways to select 5 participants out of 10.
The total number of combinations is given by:
( 5 10 ) = 5 × 4 × 3 × 2 × 1 10 × 9 × 8 × 7 × 6 = 252
Calculate the number of ways to select groups with less than 2 girls (0 or 1 girl).
0 Girls (All Boys):
( 5 6 ) = 6
1 Girl and 4 Boys:
Number of ways to choose 1 girl out of 4: ( 1 4 ) = 4
Number of ways to choose 4 boys out of 6: ( 4 6 ) = 15
Total combinations for 1 girl: 4 × 15 = 60
Calculate the number of ways to select at least 2 girls.
At least 2 girls is the total number of combinations minus the combinations with less than 2 girls:
Total combinations = 252
Combinations with less than 2 girls = 6 (0 girls) + 60 (1 girl) = 66
Combinations with at least 2 girls = 252 - 66 = 186
Calculate the probability.
The probability that the group has at least two girls is:
252 186 ≈ 0.738
Thus, Gopal found that the probability that a randomly chosen group of 5 participants has at least two girls is approximately 0.738 (rounded to three decimal places).
