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Explain what you've used and why?
There are a large number of students in Luttley College. 60% of the students are boys. Students can
choose exactly one of Games, Drama or Music on Friday afternoons. It is found that 75% of the boys
choose Games, 10% of the boys choose Drama and the remainder of the boys choose Music. Of the
girls, 30% choose Games, 55% choose Drama and the remainder choose Music.
5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
say s students
Boys = .6 s
.75 (.6) s = games
.1 (.6) s = drama
.15 (.6) s = music
Girls = .4 s
.3 (.4) s = games
.55(.4) s = drama
.15(.4) s = music
so number of drama students = .06 s + .22 s = .28 s
so if you pick one dram student
p one boy drama student in one pick = .06/.28 = .214
now do your Bernoulli thing for 1 of 5
Oh, easier find the probability that all are girls then subtract from one :)
How to know where to use conditional probability as in this question?
For all girls pretty simple
They say a large number of students so just the probability of a girl drama student to the fifth.
(.22/.28)^5 = .3
so
1 - .3 = 0.7 for one or more boys
To find the probability that at least one of the 5 randomly chosen Drama students is a boy, we need to determine the probability of the complement event, which is the probability that none of the 5 chosen students are boys and subtract it from 1.
To calculate the probability of selecting 5 Drama students with none of them being boys, we need to combine the probabilities of selecting each student one by one.
Given:
- 60% of the students are boys and 40% are girls.
- 10% of the boys choose Drama, 75% choose Games, and the rest choose Music.
- 55% of the girls choose Drama, 30% choose Games, and the rest choose Music.
To find the probability of selecting a certain type of student, we multiply the percentage of that type by the percentage of students who choose Drama.
Let's calculate the probabilities step by step:
1. Probability of selecting a boy who chooses Drama:
- Boys who choose Drama: 10% of boys = 0.10
- Probability of selecting a boy who chooses Drama = 0.10 * 0.60 (boys proportion) = 0.06
2. Probability of selecting a girl who chooses Drama:
- Girls who choose Drama: 55% of girls = 0.55
- Probability of selecting a girl who chooses Drama = 0.55 * 0.40 (girls proportion) = 0.22
Now, we need to calculate the probability of selecting 5 students who are all girls and choose Drama. To do this, we multiply the individual probabilities together:
3. Probability of selecting 5 girls who choose Drama:
- Probability of selecting a girl who chooses Drama = 0.22 (calculated in step 2)
- Probability of selecting 5 girls who choose Drama = 0.22^5 = 0.00266 (approximately)
Finally, to find the probability that at least one of the 5 selected students is a boy, we subtract the probability calculated in step 3 from 1:
Probability that at least 1 of the 5 Drama students is a boy = 1 - 0.00266 = 0.99734 (approximately)
Therefore, the probability that at least one of the 5 randomly selected Drama students is a boy is approximately 0.99734 (or 99.734%).