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.

Example 4 :

Gill rolls a number cube 78 times. How many times can he expect to roll an odd number greater than 1?

Solution :

Sample space when we roll a cube

S = {1, 2, 3, 4, 5, 6}

n(S) = 6

Let A be the event of getting odd number.

Possible outcomes for A = {3, 5}

n(A) = 2

p(A) = n(A)/n(S)

= 2/6

= 1/3

When we roll a die 78 times, 

= 78(1/3)

= 26 times.

When we roll a dies 78 times, 26 times he can expect to get an odd number greater than 1.

Example 5 :

Jenna flips two pennies 105 times. How many times can she expect both coins to come up heads?

Solution :

When we toss coins once, the sample space 

S = {HH, HT, TH, TT}

n(S) = 4

A be the event of getting heads in both coins

A = {HH}

n(A) = 1

P(A) = n(A)/n(S) 

= 1/4

When the coin is tossing 105 times,

= 105(1/4)

= 26.25

Approximately 26 times.

Example 6 :

Ron draws 16 cards from a deck of 52 cards. The deck is made up of cards of four different colors—red, blue, yellow, and green. How many of the cards drawn can Ron expect to be green?

Solution :

n(S) = 52

A deck is made up of red, blue, yellow and green.

• Number of red cards = 13
• Number of blue cards = 13
• Number of yellow cards = 13
• Number of green cards = 13

Probability of choosing a green card = 13/52

When 1 card is chosen the probability is 13/52

When 16 cards is drawn, the required probability 

= 16 x (13/52)

= 16(1/4)

= 4 cards.

Example 7 :

A bag contains 6 red and 10 black marbles. If you pick a marble from the bag, what is the probability that the marble will be black?

Solution :

Total number of marbles = 6 red + 10 black

= 16 marbles

Probability of choosing black marble = 10/16

= 5/8

Example 8 :

A bag contains 6 red and 10 black marbles. If you pick a marble, record its color, and return it to the bag 200 times, how many times can you expect to pick a black marble?

Solution :

Total number of marbles = 6 red + 10 black

= 16 marbles

Probability of selecting a black marble = 10/16

= 5/8

When the process is repeating 200 times, the expected number of getting black marble = 200 (5/8)

= 125 times.

Example 9 :

Harriet rolls a number cube. What is the probability that the number cube will land on 3 or 4?

Solution :

Sample space = {1, 2, 3, 4, 5, 6}

n(S) = 6

Probability that the number cube land on 3 or 4 = 2/6

= 1/3

Example 10 :

Harriet rolls a number cube. If he rolls the number cube 39 times, how many times can she expect to roll a 3 or 4?

Solution :

Sample space = {1, 2, 3, 4, 5, 6}

n(S) = 6

Probability that the number cube land on 3 or 4 = 2/6

= 1/3

When it is rolled 39 time, the expected number of times 

= 39 (1/3)

= 13 times

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