math question

Poisson Probabilities

The Poisson probability distribution gives the probability of a number of events

occurring in a fixed interval of time or space if these events happen with a known

average rate and independently of the time since the last event.

You will need to make sure you are looking at the correct time or interval when

you calculate Poisson probabilities. For example, if you can type, on average, 50

words in 1 minute, but you want to calculate a probability for a time of 3 minutes,

then the new average will be 50*3 = 150 words. You would use 150 to calculate

the probability NOT 50. We will do more examples below.

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 Poisson 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 Poisson Probabilities and how they would get re-written

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

• P( x = r) • P( x ≤ r) • P( x < r) = P(x ≤ r – 1) • P(x > r) = 1 – P(x ≤ r) • P( x ≥ r) same as 1 – P(x ≤ r - 1) • Expected Value is µ or 𝜆 • r is the number of occurrences

To find Poisson Probabilities we will use the =POISSON.DIST( ) function.

Example:

A Cognitive scientific designed an experiment to measure x, the number of a

times a reader’s eye fixated on a single word before moving past that word in 1

minute. They found that a person’s eye fixated on a word 3 times in 1 minute

before moving on to the next word.

1) Find the probability that a person’s eye will fixated on a word exactly 5 times in

1 minute?

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

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

P(x = 5) = POISSON.DIST(5,3, FALSE)

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

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

then the mean, 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 10.08% probability that a person’s eye will fixate on a word exactly 5

times before it moves on to the next word.

Note: When you hit “Enter” the answer will return as a decimal, .1008. You will

then need to convert it to a percent.

2) Find the probability that a person’s eye will fixate on certain words fewer than

100 times in 30 minutes?

The interval we want to look at for this problem is 30 minutes NOT 1 minute.

Because of this, we need to find the new mean. If a person’s eye fixates on a

word 3 times in 1 minute, then 3*30 = 90. Then a person’s eye will fixate on

certain words 90 times in a 30-minute time period. The new average is 90. This is

the value we will use in the Excel function.

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

This is the probability we want to find, P(x < 100)

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 < r) = P( x ≤ r – 1)

P( x < 100) = P(x ≤ 100 – 1) = P(x ≤ 99). Now that the probability is in the less than

or equal to form we can use Excel.

P(x ≤ 99) = POISSON.DIST(99,90,TRUE)

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

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

then the mean, 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 an 84.18% probability that a person’s eye will fixate on certain words

fewer than 100 times in a 30-minute period before it moves on to the next word.

Note: When you hit “Enter” the answer will return as a decimal, .8418. You will

then need to convert it to a percent.

3) Find the probability that a person’s eye will fixated on certain words at least 95

times in 30 minutes?

The time frame we are looking at is still 30 minutes so the mean will be 90.

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 ≥ 95).

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 ≥ r) = 1 - P( x ≤ r - 1)

P( x ≥ 95) = 1 - P(x ≤ 95 – 1) = 1 - P(x ≤ 94). Now that the probability is in the less

than or equal to form we can use Excel.

1 - P(x ≤ 94) = 1- POISSON.DIST(94, 90,TRUE)

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

POISSON.DIST(. From here make sure you include the left parenthesis then type

in the x value, then the mean, 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 31.28% probability that a person’s eye will fixate on certain words at

least 95 times in a 30-minute period before it moves on to the next word.

Note: When you hit “Enter” the answer will return as a decimal, .3128. You will

then need to convert it to a percent.

4) Find the probability that a person’s eye will fixated on certain words more than

97 times in 30 minutes?

The time frame we are looking at is still 30 minutes so the mean will be 90.

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

This is the probability we want to find, P(x > 97).

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 > r) = 1 – P(x ≤ r)

P( x > 97) = 1 - P(x ≤ 97)= 1 - P(x ≤ 97). Now that the probability is in the less than

or equal to form we can use Excel.

1 - P(x ≤ 97) = 1- POISSON.DIST(97, 90,TRUE)

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

POISSON.DIST(. From here make sure you include the left parenthesis then type

in the x value, the n value, then the mean, 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 21.26% probability that a person’s eye will fixate on certain words more

than 97 times in a 30-minute period before it moves on to the next word.

Note: When you hit “Enter” the answer will return as a decimal, .2126. You will

then need to convert it to a percent.