## Probability – Basic Concepts, Formulas, Tricks, Solved Problems

## Probability

Probability is a measure of the likelihood of an event to occur. Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.

Probability P (E) = Number of Wanted Outcomes / Number of Possible Outcomes

**Probability is always between 0 and 1**

### Probability Important Formulas

**Experiment** – An operation that can produce some well-defined outcomes is called an experiment.

**Random Experiment** – An experiment in which all possible outcomes are known and the exact output cannot be predicted in advance, is called a random experiment.

*Examples:*

- Rolling an unbiased dice.
- Tossing a fair coin.
- Drawing a card from a pack of well-shuffled cards.
- Picking up a ball of a certain color from a bag containing balls of different colors.

**Sample Space** – When we perform an experiment, then the set S of all possible outcomes is called the sample space.

**Probability of Occurrence of an Event**

Let S be the sample space and E be an event Then, E<= S.

- P(E) = n(E) / n(S)

**Results on Probability**

- P(S) = 1
- 0<= P(E) <= 1
- P(Ø) = 0
- For any events A and B, we have P(A U B) = P(A) + P(B) – P(A ∩ B)

### Practice Probability Aptitude Questions & Answers

**1. Tickets numbered 1 to 20 are mixed and then a ticket is randomly drawn. What is the likelihood of the ticket being drawn having a number which is a multiple of 3 or 5?**

- A. 1 / 2
- B. 2 /5
- C. 8 / 15
- D. 9 /20

**Answer:** Option D (9 /20)

**2. Three unbiased coins are tossed. What is the probability of getting at most two heads?**

- A. 3 / 4
- B. 1 / 4
- C. 3 / 8
- D. 7 / 8

**Answer:** Option D (7 / 8)

**3. There are 15 boys and 10 girls in kindergarten. Three pupils are randomly picked. Probability of choosing 1 girl and 2 boys is:**

- A. 21 / 46
- B. 25 / 117
- C. 1 / 50
- D. 3 / 25

**Answer:** Option A (21 / 46)

**4. Two cards are drawn together on random from a set of 52 cards. How is the chance that both cards will be Kings?**

- A. 1 / 15
- B. 25 / 57
- C. 35 / 256
- D. 1 / 221

**Answer:** Option D (1 / 221)

**5. One card from a pack of 52 cards is drawn at random. What is the probability that the card drawn is a face card (only Jack, Queen, and King)?**

- A. 1 / 13
- B. 3 / 13
- C. 1 / 4
- D. 9 / 52

**Answer:** Option B (3 / 13)

**6. Sam and Joan play the game of tennis. If the probability of winning Sam is 0.59 then find the probability of winning Joan.**

- A. 0.47
- B. 0.36
- C. 0.41
- D. 0.25

**Answer:** Option C (0.41)

**7. Let A and B be same sample space cases, with P (A) = 0.6 and P (B) = 0.7. Will these two things disjoint one another?**

- A. Yes
- B. No
- C. None of the above
- D. All the above

**Answer:** Option B (No)

**8. Which of these numbers is a probability not possible?**

- A. -0.00001
- B. 0.5
- C. 1.001
- D. 1

**Answer:** Option D (1)

**9. The probability of Soumya getting married within 365 days is ‘a’ and the probability of her colleague Alma getting married within 365 days is ‘b’. Consider the probability that at the end of 365 days, only one of the two will get marriages.**

- A. a-b-2ab
- B. a+b-2ab
- C. a-b+2ab
- D. ab-a-b

**Answer:** Option B (a+b-2ab)

**10.A card is drawn from a 52-card pack. The chance of being a party queen or heart king is**

- A. 1/13
- B. 2/13
- C. 1/26
- D. 1/52

**Answer:** Option C (1/26)