# How To Solve Probability Questions Quickly

## How to solve Proability Questions Quickly

Formula for Probability , P(A) = $\mathbf{ \frac{The Number Of wanted outcomes }{The total number Of Possible Outcomes}}$

Therefore,  probability of the occurence of  event is the number between 0 and 1. ### How to Solve Quickly Probability questions

• You can solve many simple probability problems just by knowing two simple rules:
• The probability of any sample point can range from 0 to 1.
• The sum of probabilities of all sample points in a sample space is equal to 1.
• The probability of event A is denoted by P(A).

### Types 1- Random ticket drawn questions.

Question 1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

Options

(a) $\frac{1}{2}$

(b) $\frac{2}{5}$

(c) $\frac{8}{15}$

(d) $\frac{9}{20}$

Solution:    Here, S = {1, 2, 3, 4, …., 19, 20}.

Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}.

P(E) = $\frac{n(E)}{n(S)}$ = $\frac{9}{20}$

Correct Options (D).

Question 2 A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

Options

(a) $\frac{10}{21}$

(b) $\frac{11}{21}$

(c) $\frac{2}{27}$

(d) $\frac{5}{7}$

Solution:    Total number of balls = (2 + 3 + 2) = 7.

Let S be the sample space.

Then, n(S) = Number of ways of drawing 2 balls out of 7.

7C2

=$\frac{7 × 6}{2 × 1}$

= 21.

Let E = Event of drawing 2 balls, none of which is blue. n(E) = Number of ways of drawing 2 balls out of (2 + 3) balls.

5C2

=$\frac{5 × 4}{2 × 1}$

= 10 .

P(E) = $\frac{n(E)}{n(S)}$ = $\frac{10}{21}$ .

Correct Options (A)

Questions 3 A bag contains 1100 tickets numbered 1, 2, 3, … 1100. If a ticket is drawn out of it at random, what is the probability that the ticket drawn has the digit 2 appearing on it?

(a) $\frac{291}{1100}$

(b) $\frac{292}{1100}$

(c) $\frac{290}{1100}$

(d) $\frac{301}{1100}$

Solution:     Within 1 to 100, numbers containing digit 2 = 19

Within 101 to 199, numbers containing digit 2 = 19

Within 201 to 300, numbers containing digit 2 = 19

Within 301 to 1100, numbers containing digit 2 = 19 × 8 =152

Probability =  $\frac{290}{1100}$ is answer.

Correct Options (c)

### Type 2- Probability questions on boys and girls

Question 4. In a class there are 60% of girls of which 25% poor. What is the probability that a poor girl is selected is leader?

Options

(a) 20%

(b) 15%

(c) 10%

(d) 25%

Solutions:   Assume total students in the class = 100

Then Girls = 60% (100) = 60

Poor girls = 25% (60) = 15

So probability that a poor girls is selected leader = $\frac{Poor girls}{ Total students}$ = $\frac{15}{100}$ = 15%

Correct Options (b)

Questions 5 What is the probability that the total of two dice will be greater than 9, given that the first die is a 5?

Options

(a) $\frac{1}{3}$

(b) $\frac{1}{6}$

(c) $\frac{1}{9}$

(d) None of these

Solution :    Let A= first die is 5

Let B = total of two dice is greater than 9

P(A) = Possible outcomes for A and B: (5, 5), (5, 6)

P(A and B) = $\frac{2}{36}$ =$\frac{1}{18}$

P(B|A) = $\frac{P(A and B)}{P(A)}$ = $\frac{1}{18}$ ÷ $\frac{1}{6}$ = $\frac{1}{3}$.

Correct Options (A)

Questions 6. If six cards are selected at random (without replacement) from a standard deck of 52 cards, what is the probability there will be no pairs? (two cards of the same denomination)

Solution:    Let Ei be the event that the first i cards have no pair among them. Then we want to compute P(E6).

Which is actually the same as P(E1 ∩ E2 ∩ · · · ∩ E6), since E6 ⊂ E5 ⊂ · · · ⊂ E1, implying that E1 ∩ E2 ∩ · · · ∩ E6 = E6.

We get P(E1 ∩ E2 ∩ · · · ∩ E6) = P(E1)P(E2|E1)· · ·  = $\frac{52}{52}$ $\frac{48}{51}$ $\frac{44}{50}$ $\frac{40}{49}$ $\frac{36}{48}$ $\frac{32}{47}$

Alternatively, one can solve the problem directly using counting techniques.

Define the sample space to be (equally likely)ordered sequences of 6 cards; then, |S| = 52 · 51 · 50 · · · 47, and the event E6 has 52 · 48 · 44 · · · 32 elements

Questions 7.  A group of 5 friends-Archie, Betty, Jerry, Moose, and Veronica-arrived at the movie theater to see a movie. Because they arrived late, their only seating option consists of 3 middle seats in the front row, an aisle seat in the front row, and an adjoining seat in the third row. If Archie, Jerry, or Moose must sit in the aisle seat while Betty and Veronica refuse to sit next to each other, how many possible seating arrangements are there?

Options

(a) 32

(b) 36

(c) 48

(d) 72

(e) 120

Solution:     Good = Total – Bad.

Total = arrangements with Archie, Jerry or Moose in the aisle seat.

Number of options for the aisle seat = 3. (Archie, Jughead, or Moose)

Number of ways to arrange the 4 other people = 4*3*2*1.

To combine these options, we multiply:  3*4*3*2 = 72.

Bad = arrangements with Archie, Jerry or Moose in the aisle seat BUT with Betty next to Veronica.

Number of options for the aisle seat = 3. (Archie, Jughead, Moose).

Number of options for the third row seat = 2. (Anyone but Betty and Veronica, since in a bad arrangement they sit next to each other.)

Number of options for the middle of the 3 remaining seats = 2. (Must be Betty or Veronica so that they sit next to each other).

Number of ways to arrange the 2 remaining people = 2*1.

To combine these options, we multiply:  3*2*2*2 = 24.

Good arrangements = 72 – 24 = 48.

Correct Option C.

Read Also –  Formulas to solve probability questions