# Probability

# Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes |

Sometimes students get mistaken for “favourable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.

## Solved Examples

**1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?**

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

**2) There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:**

- No. of blue bottles picked out: 300
- No. of red bottles: 200
- No. of green bottles: 450
- No. of orange bottles: 50

**a) What is the probability that Sumit will pick a green bottle?**

Ans: For every 1000 bottles picked out, 450 are green.

Therefore, P(green) = 450/1000 = 0.45

**b) If there are 100 bottles in the container, how many of them are likely to be green?**

Ans: The experiment implies that 450 out of 1000 bottles are green.

Therefore, out of 100 bottles, 45 are green.

## Types of Probability

There are three major types of probabilities:

- Theoretical Probability
- Experimental Probability
- Axiomatic Probability

### Theoretical Probability

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

### Experimental Probability

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and heads is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

### Axiomatic Probability

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as **Kolmogorov’s three axioms. **With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

## Probability of an Event

Assume an event E can occur in r ways out of a sum of n probable or possible **equally likely ways**. Then the probability of happening of the event or its success is expressed as;

P(E) = r/n

The probability that the event will not occur or known as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event will not occur.

Therefore, now we can say;

**P(E) + P(E’) = 1**

This means that the total of all the probabilities in any random test or experiment is equal to 1.

### What are Equally Likely Events?

When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die:

- Getting 3 and 5 on throwing a die
- Getting an even number and an odd number on a die
- Getting 1, 2 or 3 on rolling a die

are equally likely events, since the probabilities of each event are equal.

*Probability Tree*

The **tree diagram** helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagrams are used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:

PROBABILITY(questions)

*A coin is tossed 500 times and we get*

Head:285times and tails :215 times

When coin is tossed at random what is the probability of getting

- a head B. a tail?

- From a deck of cards, 10 cards are picked at random and shuffled. The cards are as follows: 6, 5, 3, 9, 7, 6, 4, 2, 8, 2 Find the probability of picking a card having value more than 5 and find the probability of picking a card with an even number on it.

**1500 families with 2 children were selected randomly, and the following data were recorded:**

Number of girls in a family | 2 | 1 | 0 |

Number of families | 475 | 814 | 211 |

Compute the probability of a family, chosen at random, having

(i) 2 girls (ii) 1 girl (iii) No girl Also,

check whether the sum of these probabilities is 1.

**A die is thrown 1000 times with the frequencies for the outcomes 1, 2, 3, 4, 5 and 6 as given in the following table :**

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 179 | 150 | 157 | 149 | 175 | 190 |

Find the probability of getting each outcome.

*Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg):*

4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00

Find the probability that any of these bags chosen at random contains more than 5 kg of flour

**The distance (in km) of 40 engineers from their residence to their place of work were found as follows: 5 3 10 20 25 11 13 7 12 31 19 10 12 17 18 11 32 17 16 2 7 9 7 8 3 5 12 15 18 3 12 14 2 9 6 15 15 7 6 12. What is the empirical probability that an engineer lives:**

(i) less than 7 km from her place of work? (ii) more than or equal to 7 km from her place of work?

(iii) within km from her place of work?

**A box contains 50 bolts and 150 nuts. On checking the box, it was found that half of the bolts and half of the nuts are rusted. If one item is chosen at random, find the probability that it is rusted.**

- A dice is rolled number of times and its outcomes are recorded as below :

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 35 | 45 | 50 | 38 | 53 | 29 |

Find the probability of getting an odd number.

**The probability of guessing the correct answer to a certain question is x/ 2.If probability of not guessing the correct answer is 2 /3 then find x.**

- The record of a weather station shows that out of the past 250 consecutive days, its weather forecast were correct 175 times.

(i) What is the probability that on a given day it was correct? (ii) What is the probability that it was not correct on a given day?

*If we throw a die, then the upper face shows 1 or 2; or 3 or 4; or 5 or 6. Suppose we throw a die 150 times and get 2 for 75 times. What is the probability of getting a �2�?*

*Twelve bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg): 4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00 5.12 Find the probability that any of these bags chosen at random contains more than 5 kg of flour*

- A four digit number is to be formed by using the digits 2, 4, 7, 8. The probability that the number will start with 7 is

- In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that on a ball played: (i) he hits boundary (ii) he does not hit a boundary.

- To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following table: Class 9 RD Sharma Solutions Chapter 25 Probability Find the probability that a student chosen at random
- likes Mathematics (ii) does not like it.

- Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg): 4.97, 5.05, 5.08, 5.03, 5.00, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00 Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

- Twenty four people had a blood test and the results are shown below. A , B , B , AB , AB , B , O , O , AB , O , B , A AB , A , O , O , AB , B , O , A , AB , O , B , A

(a) Construct a frequency distribution for the data. (b) If a person is selected randomly from the group of twenty four people, what is the probability that his/her blood type is not O?

- In the above question, Find the probability of a household having 3 mobile set and having income less than 10000

**If the probability of winning a game is 0.3, then probability of losing it is**

(a) 0.6 (b) 0.7 (c) 0.5 (d) None of these