In a binomial experiment, this spinner is spun 10 times and the result is recorded as a success if the top score is achieved. Let 𝑋 be the number of successes. Determine the probability that 𝑋 equals two as a percentage to three decimal places. Determine the probability that 𝑋 equals nine as a percentage to three decimal places.
We’re told in the question that our experiment is binomial. This means there are only two possible outcomes, success or failure. Any binomial experiment can be written in the form 𝑋 is approximately equal to the binomial of 𝑛 comma 𝑃, where 𝑛 is the number of trials, and 𝑃 is the probability of success. We’re told that the spinner is spun 10 times. Therefore, 𝑛 is equal to 10.
The experiment is said to be successful if the top score is achieved. There are eight equal sections on the spinner, two of which have the top score of 100. This means that the probability is equal to two out of eight, or two-eighths. This is equivalent to one-quarter. For the purposes of this question, we’ll use the decimal equivalent to one-quarter, which is 0.25. Our value of 𝑛 is 10. And our value of 𝑃 is 0.25.
In order to answer the two questions, the probability that 𝑋 equals two and the probability that 𝑋 equals nine, we need to recall one of our formulae. The probability that 𝑋 equals 𝑎 is equal to 𝑛 choose 𝑎 multiplied by 𝑃 to the power of 𝑎 multiplied by one minus 𝑃 to the power of 𝑛 minus 𝑎. The probability that 𝑋 equals two is, therefore, equal to 10 choose two multiplied by 0.25 squared multiplied by 0.75 to the power of eight. We get the final term as one minus 0.25 is 0.75, and 10 minus two is equal to eight.
We can type this directly into our calculator by using the 𝑛 choose 𝑟 button. This is equal to 0.2815675 and so on. As we were asked to give our answer as a percentage, we need to multiply this by 100. This moves all the digits two places to the left. We have 28.15675 and so on. We were also asked to round our answer to three decimal places. This means that the deciding number is the seven. As this is greater than five, we will round up. The probability that 𝑋 is equal to two is 28.157 percent.
We repeat this process for the second part of the question. This time, instead of the probability that 𝑋 equals two, we need to calculate the probability that 𝑋 is equal to nine. We begin with 10 choose nine. We need to multiply this by 0.25 to the power of nine. We then multiply this by 0.75 to the power of one. Typing this into the calculator gives us 0.000028610. Once again, to work out a percentage, we multiply by 100. This gives us 0.0028610 and so on. Rounding to three decimal places, the eight will be the deciding number. Once again, as this is bigger than five, we will round up. The probability that 𝑋 equals nine, written as a percentage to three decimal places, is 0.003 percent.