Chapter 15: Probability

Key Formulas:
• P(E) = Number of favorable outcomes / Total number of outcomes
• 0 ≤ P(E) ≤ 1
• P(E) + P(not E) = 1
• P(impossible event) = 0
• P(certain event) = 1

Exercise 15.1

Q1. Complete the statements:
(i) Probability of event that cannot happen is ___.
(ii) Probability of certain event is ___.
(iii) Sum of probabilities of all outcomes is ___.
(iv) Probability of impossible event is ___.
(v) Probability of an event that is sure to happen is ___.

Solution

(i) 0 (impossible event)
(ii) 1 (certain event)
(iii) 1
(iv) 0
(v) 1

Q2. Which of the following experiments have equally likely outcomes?
(i) A driver attempts to start a car. The car starts or does not start.
(ii) A player tries to shoot a target. He shoots or misses.

Solution

(i) Not equally likely - car starting is more probable
(ii) Not equally likely - shooting success depends on skill

Q3. Why is tossing a coin considered a fair way of deciding which team gets the ball?

Solution

Tossing a coin has two equally likely outcomes (H or T), each with probability 1/2. Neither team has advantage, so it is fair.

Q4. Which value of x makes P(E) = 1 impossible, if P(E) = x/(x+2)?

Solution

P(E) = 0 when x = 0
x/(x+2) = 0 → x = 0
∴ x = 0 makes it impossible

Q5. A bag contains 3 red and 5 black balls. Find probability of:
(i) red ball (ii) not red ball

Solution

Total = 3+5 = 8
(i) P(red) = 3/8
(ii) P(not red) = 5/8

Q6. A box has 5 red, 8 white, 4 green marbles. Find probability of:
(i) green (ii) not red (iii) neither white nor green

Solution

Total = 5+8+4 = 17
(i) P(green) = 4/17
(ii) P(not red) = (8+4)/17 = 12/17
(iii) P(neither white nor green) = P(red) = 5/17

Q13. A die is thrown twice. Find probability that (i) 5 will not come up either time (ii) 5 will come up at least once.

Solution

Total outcomes = 6×6 = 36
(i) P(5 not coming up) = 5/6 × 5/6 = 25/36
(ii) P(5 at least once) = 1 - 25/36 = 11/36

Q14. Two coins are tossed. Find probability of at least one head.

Solution

Outcomes: HH, HT, TH, TT
Favorable (at least one head): HH, HT, TH
P(at least one head) = 3/4

Q15. A die is thrown. Find probability of:
(i) getting a prime number
(ii) getting a number ≤ 2
(iii) getting a number > 2
(iv) getting a number not divisible by 3

Solution

(i) Prime numbers: 2, 3, 5 → P = 3/6 = 1/2
(ii) Numbers ≤ 2: 1, 2 → P = 2/6 = 1/3
(iii) Numbers > 2: 3, 4, 5, 6 → P = 4/6 = 2/3
(iv) Not divisible by 3: 1, 2, 4, 5 → P = 4/6 = 2/3

Q16. A box has 90 discs numbered 1 to 90. Find probability of:
(i) two-digit number (ii) perfect square (iii) divisible by 5

Solution

(i) Two-digit: 10-90 = 81 numbers
P = 81/90 = 9/10
(ii) Perfect squares: 1,4,9,16,25,36,49,64,81 = 9 numbers
P = 9/90 = 1/10
(iii) Divisible by 5: 5,10,...,90 = 18 numbers
P = 18/90 = 1/5

Q18. A bag has 5 red and 4 blue balls. Draw one ball at random. Find P(red).

Solution

Total = 5+4 = 9
P(red) = 5/9

Q19. A bag has 4 red, 5 black, 3 green balls. Find probability of:
(i) red (ii) not green (iii) neither red nor green

Solution

Total = 4+5+3 = 12
(i) P(red) = 4/12 = 1/3
(ii) P(not green) = (4+5)/12 = 9/12 = 3/4
(iii) P(neither red nor green) = P(black) = 5/12

Q22. A game has 10 cards numbered 1-10. Find probability of:
(i) even number (ii) perfect square

Solution

(i) Even numbers: 2,4,6,8,10 → P = 5/10 = 1/2
(ii) Perfect squares: 1,4,9 → P = 3/10

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