**Using theoretical probability to make predictions :**

We can make quantitative and qualitative predictions based on theoretical probability just as we do with experimental probability.

**Example 1 : **

A standard number cube is rolled 150 times. Predict how many times it will roll a 3 or a 4.

**Solution :**

The probability of rolling a 3 or a 4 is 2/6 = 1/3.

**Method 1 : Set up a proportion**

Write a proportion.

Write a proportion. 1 out of 3 is how many out of 150 ?

1/3 = x/150

Multiply both sides by 150.

(150)(1/3) = (x/150)(150)

150/3 = x

50 = x

**Method 2 : Set up an equation**

Multiplying probability by total number of rolls equals prediction.

(1/3) · 150 = x

50 = x

It can be expected to roll a 3 or a 4 about 50 times out of 150.

**Example 2 : **

Celia volunteers at her local animal shelter. She has an equally likely chance to be assigned to the dog, cat, bird, or reptile section. If she volunteers 24 times, about how many times should she expect to be assigned to the dog section ?

**Solution :**

The probability of being assigned to the dog section is 1/4

**Method 1 : Set up a proportion**

Write a proportion.

Write a proportion. 1 out of 4 is how many out of 24 ?

1/4 = x/24

Multiply both sides by 24.

(24)(1/4) = (x/24)(24)

24/4 = x

6 = x

**Method 2 : Set up an equation**

Multiplying probability by number of times she volunteers equals prediction.

(1/4)(24) = x

24/4 = x

6 = x

Celia can expect to be assigned to the dog section about 6 times out of 24.

**Example 3 : **

All 2,000 customers at a gym are randomly assigned a 3-digit security code that they use to access their online accounts. The codes are made up of the digits 0 through 4, and the digits can be repeated. Is it likely that fewer than 10 of the customers are issued the code 103 ?

**Solution :**

The probability of the code 103 is 1/125.

Write a proportion.

Write a proportion. 1 out of 125 is how many out of 2,000?

1/125 = x/2000

Multiply both sides by 2000.

(2000)(1/125) = (x/2000)(2000)

2000/125 = x

16 = x

It is not likely that fewer than 10 of the customers get the same code. It is more likely that 16 members get the code 103.

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