Staff, Inc., a management consulting company, is surveying...

Staff, Inc., a management consulting company, is surveying the personnel of Acme Ltd. It determined that $35 \%$ of the analysts have an MBA and that $40 \%$ of all analysts are over age $35 .$ Further, of those who have an MBA, $30 \%$ are over age 35 . a. What is the probability that a randomly chosen analyst both has an MBA and also is over age $35 ?$ b. What is the probability that a randomly chosen analyst who is over age 35 has an MBA? c. What is the probability that a randomly chosen analyst has an MBA or is over age 35 ? d. What is the probability that a randomly chosen analyst who is over age 35 does not have an MBA? e. Are the events MBA and over age 35 independent? f. Are the events MBA and over age 35 mutually, exclusive? g. Are the events MBA and over age 35 collectively exhaustive?

Staff, Inc., a management consulting company, is surveying the personnel of Acme Ltd. It determined that $35 \%$ of the analysts have an MBA and that $40 \%$ of all analysts are over age $35 .$ Further, of those who have an MBA, $30 \%$ are over age 35 .
a. What is the probability that a randomly chosen analyst both has an MBA and also is over age $35 ?$
b. What is the probability that a randomly chosen analyst who is over age 35 has an MBA?
c. What is the probability that a randomly chosen analyst has an MBA or is over age 35 ?
d. What is the probability that a randomly chosen analyst who is over age 35 does not have an MBA?
e. Are the events MBA and over age 35 independent?
f. Are the events MBA and over age 35 mutually, exclusive?
g. Are the events MBA and over age 35 collectively exhaustive?

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time, meaning they have no overlap; hence, P(A ? B) is zero. This concept is important when determining the probability of either event occurring, as there is no scenario where they both happen together.

Collectively Exhaustive Events

Collectively exhaustive events are a set of events that cover the entire sample space, meaning that at least one of the events must occur. Their probabilities sum to 1. This concept ensures that all possible outcomes are accounted for in a probabilistic model.

Independent Events

Events are independent if the occurrence of one does not affect the probability of the other. Mathematically, A and B are independent if and only if P(A ? B) = P(A) × P(B). This property simplifies calculations and is a fundamental concept in probability when analyzing events that do not influence each other.

Complementary Events

Complementary events are a pair of events where one event occurs if and only if the other does not. The probability of a complementary event is given by 1 minus the probability of the event, i.e., P(Not A) = 1 ? P(A). This concept is useful when it is easier to calculate the probability of an event not occurring.

Union of Events

The union of two events is the event that at least one of the events occurs. The probability of the union is calculated using the addition rule: P(A ? B) = P(A) + P(B) ? P(A ? B). This rule ensures that probabilities common to both events are not counted twice, maintaining correct probability evaluation.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated as P(B|A) = P(A ? B) / P(A), assuming P(A) is not zero. This concept is crucial for understanding how the occurrence of one event can influence the likelihood of another.

Intersection of Events

This concept refers to the probability that two events occur simultaneously. It is computed by using the formula P(A ? B) = P(A) multiplied by the conditional probability of B given A, or vice versa. This idea is essential in scenarios where you want to know the chance of both events happening at the same time.

Transcript

00:01 So we're told that the probability that someone has an mba is equal to 0 .35 % the probability that someone is over 35 is 0 .4 or 40%.

00:17 And the probability, we have a conditional, given that someone has an mba, that they're over 35, that probability is 0 .3.

00:28 And on the first question we want to find the probability that someone has an mba and they're over 35.

00:40 And we have two ways to find that.

00:42 We can find the probability of an mba times the probability of over 35 given that someone has an mba.

00:51 Or we could find the probability of the over 35 first times the probability of having an mba conditional on the first condition happening.

01:06 And we have the information for this one.

01:09 So if we take the 0 .35 times the 0 .4, excuse me, the point three, that product will end up giving us our intersection.

01:24 And so 0 .35 times 0 .3 comes up with 0 .105.

01:29 Now, we could draw a venn diagram of this at this point, and this could be the mba, and this could be the being over 35, and the intersection right here is at 0 .105.

01:43 And then these two together would add up to 0 .35.

01:47 We don't have to do this, but sometimes just to make the question a little more understanding, it is helpful.

01:55 So this area here would be 0 .245, and then these two together have to add up to that .4, and 0 .4 minus .105 would give us this .295.

02:11 So we don't need to do that, but again, it's helpful.

02:15 So part b, part b, we want to find the probability of having an mda, given that you know the person is over 35.

02:28 So we need the probability of the intersection, the mda, and, or intersection of over 35, divided by the probability of being over 35.

02:44 And so we know the intersection we just found is at 0 .105.

02:49 So if we must have found that, this would also be wrong.

02:51 And then the probability of somebody being over 35 is 0 .4, which means picture -wise, we're taking this intersection.

02:59 Divided by the whole circle and the 0 .105 divided by 0 .4 comes out to be 0 .2625.

03:11 So those are the first two.

03:14 Now part c, part c we want to find the probability that someone has an mba or they're over 35.

03:27 And formula -wise, that says to take the sum of the two minus the intersection, we could have a also just add these three little regions together.

03:34 So you could do it either way if you had that bend diagram.

03:36 So let's just take the 0 .35, the traditional way, plus the point four, and then subtract away the overlap, because that had been double counted...

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