# How to use the PROB function

**What is the PROB function?**

The PROB function calculates the probability that values in a range are between a given lower and upper limit.

#### Table of Contents

## 1. Introduction

**What is probability?**

Probability is how likely an event will occur, it quantifies the chance/risk of something happening.

- Probability values range from 0 (impossible) to 1 (certain).
- Probabilities of all possible outcomes sum to 1.
- Probabilities can be expressed as fractions, decimals or percentages.

Fraction example: 1/7

Decimal example: 0.1428

Percentage example: 14.28% - Probability models randomness and uncertainty.

## 2. PROB Function Syntax

PROB(*x_range, prob_range, [lower_limit], [upper_limit]*)

## 3. PROB Function Arguments

X_range |
Required. The bond's settlement date, in other words, the date a buyer purchases a security. |

Prob_range |
Required. The bond's maturity date, in other words, when it expires. |

Lower_limit |
Optional. The lower bound on the range for which you want to know the probability. |

Upper_limit |
Optional. The upper bound on the range for which you want to know the probability. If omitted the PROB function returns the probability of being equal to the lower_limit. |

## 4. PROB Function Example 1

**Two fair dice are rolled, and the sum of the numbers on the two dice is recorded. What is the probability that the sum of the two dice falls between 5 and 9 (inclusive)?**

This example demonstrates how likely it is for two dice to sum up to given outcomes. This example uses two dice with 6 sides numbered from 1 to 6. Each die has 6 possible outcomes: 1, 2, 3, 4, 5, and 6.

To solve this question, we need to consider the possible outcomes when rolling two dice and their respective probabilities. The table in cell range B19:F25 shows the outcome of every possibility.

For example, to get the total two you need the first die to show 1 and the second die also to show 1. The table shows that this outcome is not very likely because only one of 36 outcomes has the sum two.

On the other hand, seven is the most likely outcome because there are six different outcomes with the same total of seven. To calculate the probabilities of each outcome I use this formula in cell C3:

The COUNTIF function counts the number of outcomes based on the corresponding value in cell range B3:B13. It then divides the count by the total number of outcomes which is 36 in this example. We now have a table we can use to calculate the probability of getting a total of 5 (lower limit) up to 9 which is the upper limit.

Formula in cell C15:

The result is a probability of 66.67% of getting a total between 5 and 9 using two dice. The probability of getting:

- 5 is 11.11%
- 6 is 13.89%
- 7 is 16.67%
- 8 is 13.89%
- 9 is 11.11%

The total is 11.11 + 13.89 + 16.67 + 13.89 + 11.11 = 66.67%

## 5. PROB Function Example 2

**A scientist has observed 8 different events and their corresponding probabilities for them to happen. Here is the data:**

Data |
Probability |

0 | 10.00% |

1 | 5.00% |

2 | 40.00% |

3 | 14.00% |

4 | 3.00% |

5 | 14.00% |

6 | 5.00% |

7 | 9.00% |

**What is the probability that event 1 or 2 , or 3 happens?**

The events and corresponding probabilities are in cell range B3:C10. The lower limit is specified in cell C12 and the upper limit is in cell C13.

Here are the arguments:

- x_range = B3:B10
- prob_range = C3:C10
- [lower_limit] = 1
- [upper_limit] = 3

Formula in cell C15:

The function returns a probability value of 59%. Event 1 has a probability of 5%, 2 has 40% and 3 has 14%. The total is 5+40+14 = 59%

## 6. PROB Function Example 3

**What is the probability to draw a jack or a queen or a king from a standard deck of cards (52 cards)?**

The total number of possible outcomes (all cards in the deck) is 52. Each card has an equal probability of being drawn (1/52). There are 4 jacks, queens and kings each.

We can use the PROB function to calculate the probability of getting a knight, queen, or king in a single draw from the deck. 11, 12, and 13 represent jacks, queens, and kings respectively, their corresponding probabilities are 7.69%. (4/52)

The arguments are:

- x_range = B3:B15
- prob_range = C3:C15
- [lower_limit] = 11
- [upper_limit] = 13

The formula in cell C21 calculates the likelihood of drawing a face card, specifically a jack, queen, or king, when selecting a single card from a standard deck of playing cards.

Formula in cell C21:

The result is a probability value of 23.08%, 7.69 + 7.69 + 7.69 =23.08%

12/52 = 23.08%

## 7. PROB Function not working

The PROB function returns

- #NUM! error of if
- a value in
*Prob_range*is equal or smaller than 0 (zero) or larger than 1. - if the sum in
*Prob_range*is not equal to 1.

- a value in
- #N/A! error if there are a different number of values in
*Prob_range*and*X_range.*

## 8. How is the PROB Function calculated?

PROB function calculates the cumulative probability for a distribution by summing the individual probabilities between the limits. The PROB function does this by adding the percentages from the table based on the given lower and upper limits.

P(lower_limit ≤ X ≤ upper_limit)

Where:

P = Probability

X = Random variable

lower_limit = Lower bound

upper_limit = Upper bound

This calculates the cumulative probability of X being between the lower and upper limits.The PROB function implements this by:

Looking up the value (x) in the provided x_range and getting the corresponding probability (p) from the prob_range. Summing all the probability values p where x is between the lower and upper limits.

Mathematically, this is: Σ p(x) for all x where lower_limit ≤ x ≤ upper_limit

### Functions in 'Statistical' category

The PROB function function is one of 73 functions in the 'Statistical' category.

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