math question

Binomial Probabilities

The Probability of an event can be expressed as a binomial probability if its

outcomes can be broken down into two probabilities, p, which is a success and a

q, which is a failure. Where p and q are complementary

p + q = 1, thus q = 1 - p

You need to rewrite the probabilities in the less than or equal to form to use the

function in EXCEL. We will use Excel to find Binomial Probabilities. The

probabilities do need to be in the less than or equal to form to use Excel. This is

very important.

Here are some common Binomial Probabilities and how they would get re-written

to calculate in the less than or equal to form, to use Excel.

• P( x = j) • P( x ≤ j) • P( x < j) = P(x ≤ j – 1) • P(x > j) = 1 – P(x ≤ j) • P( x ≥ j) same as 1 – P(x ≤ j - 1) • Expected Value = n*p

• Standard deviation = √𝑛 ∗ 𝑝 ∗ 𝑞

• Recall, “n” denotes the sample size To find Binomial Probabilities we will use the =BINOM.DIST( ) function.

Let’s use our Car Price Data from Week 2 and calculate 4 different probabilities

Car Price: Observation 1 $ 20,000

Observation 2 $ 25,000

Observation 3 $ 30,000 Observation 4 $ 31,000

Observation 5 $ 22,500 Observation 6 $ 25,000

Observation 7 $ 29,500

Observation 8 $ 24,000 Observation 9 $ 24,500

Observation 10 $ 25,000

1) Using our data, we found the average = $25,650.

Looking at our data we see that 7 out of 10 cars fall below the average. We will

call this a success if your price falls below the average. This means p = 7/10 = .70

and q = 1 - .70 = .30.

If you were to find another random sample of 10 cars based on the same data,

what is the probability that exactly 5 of them will fall below the average?

Because of the word “exactly” we want to find this probability P(x = 5). We will

use the BINOM.DIST() function to find this probability.

P(x = 5) = BINOM.DIST(5, 10, .70, FALSE)

In Excel make sure you hit the “=“ sign first then start typing in BINOM.DIST(

From here make sure you include the left parenthesis then type in the x value,

the n value, the p value (the probability), then either TRUE or FALSE. Then close

the parenthesis ) and hit Enter.

Type in a TRUE when you have a less than or equal to probability and type in a

FALSE when you have an equals probability. This example has an “=“ sign so we

will use a FALSE.

There is a 10.29% probability that exactly 5 of the cars will fall below the average.

Note: When you hit “Enter” the answer will return as a decimal, .1029. You will

then need to convert it to a percent.

2) If you were to find another random sample of 10 cars based on the same data,

what is the probability that fewer than 8 of them will fall below the average?

Because of the word “fewer” we will use the less than sign.

This is the probability we want to find, P(x < 8)

This probability is in the less than form NOT the less than or equal to form so we

need to rewrite this in the less than or equal to form.

Remember: P( x < j) = P( x ≤ j – 1)

P( x < 8) = P(x ≤ 8 – 1) = P(x ≤ 7). Now that the probability is in the less than or

equal to form we can use Excel.

P(x ≤ 7) = BINOM.DIST(7,10,.70,TRUE)

In Excel make sure you hit the “=“ sign first then start typing in BINOM.DIST(.

From here make sure you include the left parenthesis then type in the x value,

the n value, the p value (the probability), then either TRUE or FALSE. Then close

the parenthesis ) and hit Enter.

Type in a TRUE when you have a less than or equal to probability and type in a

FALSE when you have an equals probability. This example has an “≤“ sign so we

will use a TRUE.

There is a 61.72% probability that fewer than 8 of the cars will fall below the

average.

Note: When you hit “Enter” the answer will return as a decimal, .6172. You will

then need to convert it to a percent.

3) If you were to find another random sample of 10 cars based on the same data,

what is the probability that at least 3 of them will fall below the average?

Because of the words “at least” we will use the greater than or equal to sign.

This is the probability we want to find, P(x ≥ 3).

This probability is in the greater than or equal to form NOT the less than or

equal to form so we need to rewrite this in the less than or equal to form.

Remember: P( x ≥ j) = 1 - P( x ≤ j - 1)

P( x ≥ 3) = 1 - P(x ≤ 3 – 1) = 1 - P(x ≤ 2). Now that the probability is in the less than

or equal to form we can use Excel.

1 - P(x ≤ 2) = 1- BINOM.DIST(2,10,.70,TRUE)

In Excel make sure you hit the “=“ sign first, then the 1 - and then, BINOM.DIST(.

From here make sure you include the left parenthesis then type in the x value,

the n value, the p value (the probability), then either TRUE or FALSE. Then close

the parenthesis ) and hit Enter.

Type in a TRUE when you have a less than or equal to probability and type in a

FALSE when you have an equals probability. This example has an “≤“ sign so we

will use a TRUE.

There is a 99.84% probability that at least 3 of the cars will fall below the average.

Note: When you hit “Enter” the answer will return as a decimal, .9984. You will

then need to convert it to a percent.

4) If you were to find another random sample of 10 cars based on the same data,

what is the probability that more than 5 of them will fall below the average?

Because of the word “more” we will use the greater than sign.

This is the probability we want to find, P(x > 5).

This probability is in the greater than form NOT the less than or equal to form so

we need to rewrite this in the less than or equal to form.

Remember: P(x > j) = 1 – P(x ≤ j)

P( x > 5) = 1 - P(x ≤ 5)= 1 - P(x ≤ 5). Now that the probability is in the less than or

equal to form we can use Excel.

1 - P(x ≤ 5) = 1- BINOM.DIST(5,10,.70,TRUE)

In Excel make sure you hit the “=“ sign first, then the 1 - and then, BINOM.DIST(.

From here make sure you include the left parenthesis then type in the x value,

the n value, the p value (the probability), then either TRUE or FALSE. Then close

the parenthesis ) and hit Enter.

Type in a TRUE when you have a less than or equal to probability and type in a

FALSE when you have an equals probability. This example has an “≤“ sign so we

will use a TRUE.

There is a 84.97% probability that more than 5 of the cars will fall below the

average.

Note: When you hit “Enter” the answer will return as a decimal, .8497. You will

then need to convert it to a percent.