Question

7. A fair spinner is made up of 8 sectors of the same size. If the spinner is spun once, determine the probability

of each of the following events. [4 points]
(a) The spinner landed on a prime number.

(b) The spinner landed on a 7 and a number less than 4.

(c) The spinner landed on an even or

(d) The spinner landed on a 4 or a multiple of 2.

Answer 1

a) There are 4 prime numbers less than or equal to 8: 2, 3, 5, and 7. Therefore, the probability of the spinner landing on a prime number is 4/8 or 1/2.

(b) The only sector that satisfies this condition is the sector labeled 3. Therefore, the probability of the spinner landing on a 7 and a number less than 4 is 1/8.

(c) Half of the sectors are even numbers (2, 4, 6, and 8), so the probability of the spinner landing on an even number is 4/8 or 1/2.

(d) The sector labeled 4 satisfies this condition, as well as the sectors labeled 2, 6, and 8 (which are all even numbers). Therefore, the probability of the spinner landing on a 4 or a multiple of 2 is 4/8 or 1/2.

Answer 2

(a) 1/2

(b) 0

(c) 5/8

(d) 1/2

Explanation:

The probability of an event happening is calculated by dividing the number of ways that the event can occur by the total number of possible outcomes.

`\boxed{\textsf{Probability of an event happening}=\frac{\textsf{Number of ways the event can occur}}{\textsf{Total number of possible outcomes}}}`

As the spinner is fair and is made up of 8 sectors of the same size, the total number of possible outcomes is 8.

Part (a)

A prime number is a positive integer greater than 1 that is divisible by only 1 and itself. So, the primes numbers between 1 and 8 are 2, 3, 5 and 7. Therefore, there are 4 ways the spinner can land on a prime number.

Since the spinner is fair and has 8 sectors of equal size, each prime number sector has the same probability of being landed on. Therefore, the probability of landing on a prime number is:

`\textsf{P(prime)}=(4)/(8)=(1)/(2)`

Part (b)

As the spinner is spun only once, and 7 is not less than 4, there is no way to satisfy this condition. Therefore, the probability of the spinner landing on a 7 and a number less than 4 is zero:

`\textsf{P(7 and a number less than 4)}=(0)/(8)=0`

Part (c)

An even number is a whole number that is divisible by 2. So, the even numbers between 1 and 8 are 2, 4, 6, and 8.

Therefore, the probability of landing on an even number is:

`\textsf{P(even number)}=(4)/(8)`

There is one sector labeled 5.

Therefore, the probability of landing on a 5 is:

`\textsf{P(landing on 5)}=(1)/(8)`

As 5 is not an even number the events are mutually exclusive.

Therefore, the probability of landing on an even number or a 5 can be calculated by adding the probabilities of each event:

`\textsf{P(even number or 5)}=(4)/(8)+(1)/(8)=(5)/(8)`

Part (d)

The multiples of 2 between 1 and 8 are 2, 4, 6 and 8.

As 4 is a multiple of 2, the events are not mutually exclusive.

Therefore, the probability of landing on a 4 or a multiple of 2 is:

`\textsf{P(4 or multiple of 2)}=(4)/(8)=(1)/(2)`