A binomial distribution is used in probability theory and statistics. As the basis for the binomial test of statistical significance, binomial distributions are typically used to model the number of successful events in success/failure experiments. The three assumptions underlying the distributions are that each trial has the same probability of occurring, there can only be one outcome for each trial, and each trial is a mutually exclusive independent event.

Binomial tables can sometimes be used to calculate probabilities instead of using the binomial distribution formula. The number of trials (n) is given in the first column. The number of successful events (k) is given in the second column. The probability of success in each individual trial (p) is given in the first row at the top of the table.

## The Probability of Choosing Two Red Balls in 10 Tries

Evaluate the probability of choosing two red balls out of 10 tries if the probability of choosing a red ball equals 0.2.

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Begin at the upper left corner of the binomial table at n = 2 in the first column of the table. Follow the numbers down to 10 for the number of trials, n =10. This represents 10 tries to obtain the two red balls.

Locate k, the number of successes. Here success is defined as choosing two red balls in 10 tries. In the second column of the table, find the number two representing successfully choosing two red balls. Circle the number two in the second column and draw a line under the entire row.

Return to the top of the table and locate the probability (p) in the first row across the top of the table. The probabilities are given in decimal form.

Locate the probability of 0.20 as the probability a red ball will be chosen. Follow down the column under 0.20 to the line drawn under the row for k = 2 successful choices. At the point that p = 0.20 intersects k = 2 the value is 0.3020. Thus, the probability of choosing two red balls in 10 tries equals 0.3020.

Erase the lines drawn on the table.

## The Probability of Choosing Three Apples in 10 Tries

Evaluate the probability of choosing three apples out of 10 tries if the probability of choosing an apple = 0.15.

Begin at the upper left corner of the binomial table at n = 2 in the first column of the table. Follow the numbers down to 10 for the number of trials, n =10. This represents 10 tries to obtain the three apples.

Locate k, the number of successes. Here success is defined as choosing three apples in 10 tries. In the second column of the table, find the number three representing successfully choosing an apple three times. Circle the number three in the second column and draw a line under the entire row.

Return to the top of the table and locate the probability (p) in the first row across the top of the table.

Locate the probability of 0.15 as the probability an apple will be selected. Follow down the column under 0.15 to the line drawn under the row for k = 3 successful choices. At the point where p = 0.15 intersects k = 3 the value is 0.1298. Thus, the probability of choosing three apples in 10 tries equals 0.1298.