Question

Of 180 students in a massive group, 135 are scholarship holders, 146 dedicate part of their time to work; and 114 are scholarship holders and dedicate part of their time to work. If a student is selected at random, what is the probability that: a) The chosen student is awarded a scholarship or dedicates part of his time to work b) The selected student is not awarded a scholarship and does not spend part of his time working c) A sample of 6 students was chosen, which is the probability that the sample will have at least 5 scholars. (Here only consider two groups, the scholars and the non-scholars)

Answer 1

P(S) = 135/180

P( W) = 146/180

P( S and W ) = 114/180

where S = scholar

W = dedicate part of time to work

a) To find P( S or W)

We know that , P( S or W) = P( S) +P(W) -P( S and W)

= 135/180+ 146/180- 114/180

= 0.9278

Probability that the selected student is a scholar or dedicates part of time to work = 0.9278

b) To find P( S' and W' )

where , S'= non scholar

W' = do not dedicate part of time to work

P(W) = P( S and W) + P( S' and W)

thus, P( S' and W) = P( W) - P( S and W) = 146/180 - 114/180 = 0.1778

P(S') = 1- P(S)=1-135/180 = 0.25

P(S') = P( S' and W ) + P( S' and W')

then , P( S' and W') = P(S') - P( S' and W) = 0.25 - 0.1778 = 0.0722

Probability that selected student does not have a scholarship and does not dedicate part of time to work = 0.0722

(c) P(S) = 135/180 = 0.75

which is the probability that a person is scholar , let p = 0.75

Let X be the number of scholars in sample of 6

X follow Binomial with n= 6 , p =0.75

The probability mass function of X is

= 0.5339

Probability that sample will have at least 5 scholars = 0.5339