Board of Directors The University of Colorado Board of Directo...

Question

Board of Directors The University of Colorado Board of Directors has 23 members. One member serves as board chair and another serves as vice chair. Given the names of the 23

board members, what is the probability of randomly selecting the name of the chair and the name of the vice chair? (Source: University of Colorado)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A student council has 18 members. One member is chosen as president and another as secretary. If the names of all 18 members are placed in a hat, what is the probability of randomly selecting the name of the president and the name of the secretary? Awesome. So it appears for this particular problem we're ultimately trying to determine what the probability is of randomly selecting the name of the president and the name of the secretary. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. So A is 1 divided by 272, B is 1 divided by 324. C is 1 divided by 306, and D is 1 divided by 153. Awesome. So first off, let us recall a note that there are 18 choices for president, and after the president is chosen, there are 17 remaining choices for the secretary. And in order to solve this particular problem, we will need to recall and use the fundamental counting principle, which states, as we should recall, that if one event can occur in M ways and another event can occur in N ways, then the two events can occur together in M multiplied by N ways. Therefore, the number of ways to choose both is 18 multiplied by 17, which is equal to 306. And as we should recall a note, if we pick two names at random, the order matters since one is president and one is secretary. Therefore, as a result, this is the same as counting the number of ways to select an ordered pair from 18 names. So with that in mind, we need to recall and use the concept of a permutation, and as we should recall, a permutation is an ordered arrangement of objects. The number of permutations of distinct objects taken R at a time is P. N R is equal to N factoral divided by parentheses N minus R multiplied by factoral. So that will mean that P 18.2, which you might hear this as 18 chooses 2, so the probability of 18 chooses 2, so P of 18.2. is equal to 18 factoral divided by parentheses 18 minus 2 multiplied by factoral, which when we plug that into our calculator, we will find that this is equal to 306. Since both approaches give us 306 ways, because we are picking and arranging the people for a specific roles, is considered a permutation. And the fundamental counting principle is the basis for the permutation formula. And we also need to note and recall that there is only one specific way to get a pair for one president and for one secretary. So therefore, without a doubt, the probability of randomly picking the president and secretary as a specific pair is 1 divided by 306, and that is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter C, which once again is 1 divided by 306. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Probability

Combinations

Random Selection

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