a We know that one student is to be chosen to present the Best Teacher award at a school banquet. We want to describe the process that gives us a "fair" chance to be chosen.
Fair means equally likely.
We want to give every student from a group, including you, the same chance to be chosen. Therefore, instead of looking at the number of hours spent on preparation, we are going to consider each student as an equal individual.
The only criteria of choice will be the number of possiblechoices — the number of students in the group. Let's find it!
1+19=20
In order to carry out the choosing process, we can prepare 20 draws. Each draw will be a piece of paper, on which the name of one student from the group will be written. Later, we are going to put the draws into the box.
Now we can perform the drawing. This process guarantees that every student is equally likely to be chosen because there is exactly one paper in the box with the name of each student. Next, we want to find the probability that you will be the one chosen. Let's recall the Theoretical Probability Formula.
P=PossibleOutcomesFavorableOutcomes
Since there is only 1 draw with your name on it, the number of favorableoutcomes is 1. We have already found that the number of all students in the group is 20. This is the number of possibleoutcomes. We are ready to find the desired probability.
The probability that you are chosen to present the award is 5%. Keep in mind that this is just one possibility for creating a fair process, and that there are other possible methods.
b This time, we have to consider a different definition of a fair choice.
Fair means proportional to the number of hours each student worked in preparation.
We know the whole group worked a combined total number of 45 hours during the preparation. In order to make the choice of the presenter proportional to the number of hours, in this case we will prepare 45 cards to later write names on them.
We will write your name not on one card, like in Part A, but on 4 cards. This is because you worked 4 hours on your presentation and we want the number of cards — and through that the probability of being chosen — to be proportional to the number of hours spent on preparation.
The rest of the cards will have the names of other students from the group written on them. The number of cards will correspond to the number of hours each of them has worked on preparation, so it may vary. Now, let's use the theoretical probability to find the probability that you will be the one chosen to present the award.
P=PossibleOutcomesFavorableOutcomes
Since there are 4 cards with your name on it, the number of favorableoutcomes is 4. We have prepared a total of 45 cards, which is the number of possibleoutcomes. With this, we have enough information to find the desired probability.
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