Video: Selecting the Sample Space for an Event

Which of the following represents the sample space of choosing at random a prime number between 3 and 59? [A] 𝑆 = {5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} [B] 𝑆 = {7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43} [C] 𝑆 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59} [D] 𝑆 = {5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43} [E] 𝑆 = {7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}.

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Video Transcript

Which of the following represents the sample space of choosing at random a prime number between three and 59? Is it set A, set B, set C, set D, or set E?

We can immediately rule out set C as it contains numbers that are not between three and 59. Two, three, and 59 are not between three and 59. Let’s now consider all the numbers between three and 59. This consists of all the numbers from four to 58 inclusive.

We need to find all the prime numbers in this list. Well, a prime number has only two factors: one and itself. Two is the only even prime number. Therefore, we can cross out any number that is even.

Now let’s look for any numbers in the three times table. Any number that is divisible by three cannot be prime, as it would have more than two factors. Some of these numbers have already been crossed out, for example, six and 12. But we can also eliminate nine, 15, 21, 27, 33, 39, 45, 51, and 57. As we have already crossed out all the even numbers, we don’t need to worry about the four, six, eight, or any other even times table.

Let’s now look for numbers that are in the five times table. Well, most of these have already been crossed out. But we can also cross out 25, 35, and 55. We didn’t cross out the number five as five is a prime number, because it only has two factors: one and itself.

Now let’s move on to the seven times table. Seven is a prime number, as again it is only divisible by one and itself. It has two factors. Are there any other numbers in the seven times table that we haven’t crossed out? Well, the only number remaining in the seven times table that needed to be crossed out was 49. 49 cannot be prime as it is divisible by seven.

We can continue this process with the 11 times table. However, there are no numbers in the 11 times table — 22, 33, 44, 55 — that have now already been crossed out. In fact, all the numbers that are remaining are prime numbers as they are all divisible only by one and itself. They only have two factors.

53 only has factors one and 53. 47 only has factors one and 47. The same applies to numbers 13, 17, 19, 23, 29, 31, 37, 41, and 43. This means that the sample space of choosing a random prime number between three and 59 consists of the set of numbers five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and 53.

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