Question

According to CalCentral, out of the 297 people in our Stat 2 class, 71 have senior standing (7 or more semesters at UCB), and 226 do not. Among those with senior standing, 53 have declared a major and 18 have not. Among those without senior standing, 69 have declared a major and 157 have not. Suppose we are going to choose a single student at random from the class.




(a) Are the events “drawing someone with senior standing” and “drawing someone who hasn’t declared a major” mutually exclusive?



YES or NO (circle one) and justify your answer:









(b) Are the events “drawing someone with senior standing” and “drawing someone who hasn’t declared a major” independent?



YES or NO (circle one) and justify your answer:










Stat 2, Spring 2020 Midterm 2 Name____________________
 SID______________________

Now suppose we are going to draw five students at random without replacement. In answering the following questions, express your answers as percentages rounded to the nearest tenth of a percent.






(c) What is the conditional probability that the second student drawn has senior standing, if we know the first student drawn has senior standing?





Answer: _____________ 





(d) What is the probability that the second student drawn has senior standing, not knowing anything about the first?





Answer: _____________





(e) What is the probability that among the five students, none have senior standing?





Answer: _____________


Answer 1

Let S= Event that people have Senior Standing   M = Event that people have declared a major S' = Event that people does not have Senior Standing M' = Event that people have not declared a major

Given: Total No. of Sample = 297, n(S) = 71, n(S') = 226, n(SM) = 53, n(SM') = 18, n(S'M) = 69, n(S'M)' = 157 where the complementary event S' and M' are the complementary probabilities.

(a) Events of “drawing someone with senior standing” and “drawing someone who hasn’t declared a major”are said to be mutually exclusive if probability of both events occurring together is zero (If occurance of one event prevents the occurance of the other). But the intersection of the probabilities of the two events:

Hence, we cannot conclude that the two events are mutually exclusive. '

Ans. NO

(b) (b) Events “drawing someone with senior standing” and “drawing someone who hasn’t declared a major” are said to be independent of the intersection of their probabilities P(SM') is equal to the product of their individual probabilities (P(S) and P(M').

Here, P(SM') = 0.06 .............................(1)

...................................(2)

We find that the two probabilities are not equal. Hence,

Ans. NO

(c) On drawing five students at random:

Conditional probability that the second student drawn has senior standing, if we know the first student drawn has senior standing:

Here, first student selected already has a senior standing (one student drawn from 71 out of 297) and once the first student is drawn, the next with senior standing is drawn from 71 - 1 =70 students out of 297 - 1 = 296 students.

Required probability

= 0.057

(d) The probability that the second student drawn has senior standing, not knowing anything about the first:

The first student selected might be either the one with Senior standing or not.Hence, the required probability would include the probability of first case, where, student drawn has senior standing or the first case where, he does not have senior standing.

Required probability

= 0.057+ 0.183

= 0.239

(e) Probability that among the five students, none have senior standing:

Required probability

= 4699109520 / 18616750614

= 0.252