What are the chances of getting a diverse subcommittee?

• BrownianMan
It's basically the probability of getting a red ball from each urn and then adding them together. And for the second part, it's finding the probability of getting heads and a red ball, then dividing it by the probability of getting a red ball. So yes, your calculations look correct.
BrownianMan
1. A committee consists of five Chicanos, two Asians, three African Americans, and two Caucasians. A subcommittee of four is chosen at random. What is the probability that all the ethnic groups are represented on the subcommittee?

I know the answer for this is (5x3x2x2)/(12 choose 4), but I'm not sure exactly why. I understand why the denominator is 12 choose 4), but I can't figure out why the numerator is (5x3x2x2).

2. Urn A has three red balls and two white balls, and urn B has two red balls and five white balls. A fair coin is tossed; if it lands heads up, a ball is drawn from urn A, and otherwise, a ball is drawn from urn B. What is the probability that a red ball is drawn? If a red ball is drawn, what is the probability that the coin landed heads up?

Ok, so for this I got,

P(red ball is drawn) = P(heads)P(red ball drawn from urn A) + P(tails)P(red ball drawn from urn B) = (1/2)(3/5) + (1/2)(2/7) = 31/70

then,

P(heads given red ball drawn) = P(heads and red ball drawn)/P(red ball drawn) = (1/2)(3/5)/(31/70) = 21/31

Is this right?

1) The numerator is the number of ways you can get one person of each ethnicity if you pick 4 people. How many ways can you pick four people so that one person is Chicano, one is Asian, one is African American, and one is Caucasian? There are 5 choices for a Chicano. For each Chicano you can pick one of 2 Asians, so there are 5x2 ways to pick one Chicano and one Asian. Extend this to picking African Americans and Caucasians and you get 5x2x3x3.

2) I think that's right.

What is the difference between independent and dependent events?

Independent events are events that do not affect each other's probability. This means that the outcome of one event does not depend on the outcome of the other event. Dependent events, on the other hand, are events that do affect each other's probability. This means that the outcome of one event depends on the outcome of the other event.

How do you calculate the probability of two independent events occurring?

To calculate the probability of two independent events occurring, you multiply the probabilities of each event. For example, if the probability of event A is 0.5 and the probability of event B is 0.3, then the probability of both events occurring is 0.5 x 0.3 = 0.15 or 15%.

What is the difference between mutually exclusive and non-mutually exclusive events?

Mutually exclusive events are events that cannot occur at the same time. This means that if one event happens, the other event cannot happen. Non-mutually exclusive events, on the other hand, are events that can occur at the same time. This means that both events can happen simultaneously.

How do you calculate the probability of two dependent events occurring?

To calculate the probability of two dependent events occurring, you multiply the probability of the first event by the probability of the second event, given that the first event has occurred. This can be written as P(A and B) = P(A) x P(B|A), where P(B|A) is the conditional probability of event B given that event A has occurred.

What is the difference between a sample space and an event space?

The sample space is the set of all possible outcomes of an experiment. It includes all possible outcomes, even if they have a probability of 0. The event space, on the other hand, is the subset of the sample space that includes all the outcomes that satisfy a given condition. It only includes outcomes that have a non-zero probability of occurring.

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