Exercise 14.1: Theoretical probability
What it tests. List the equally likely outcomes, count the favourable ones, divide. \(0 \le P(E) \le 1\); \(P(\text{not } E) = 1 - P(E)\). A deck has 52 cards: 4 suits of 13, 26 red, 12 face cards (J, Q, K).
Exercise 14.1, Question 1
Complete the statements.
(i) \(P(E) + P(\text{not } E) =\) ____
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- An event either happens or it doesn't.
Answer: \(1\)
(ii) The probability of an event that cannot happen is ____; such an event is called ____.
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- No outcome is favourable.
Answer: \(0\); an impossible event
(iii) The probability of an event that is certain to happen is ____; such an event is called ____.
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- Every outcome is favourable.
Answer: \(1\); a sure (certain) event
(iv) The sum of the probabilities of all the elementary events of an experiment is ____.
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- Together they cover every outcome exactly once.
Answer: \(1\)
(v) The probability of an event is greater than or equal to ____ and less than or equal to ____.
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- Favourable outcomes are between none and all of them.
Answer: \(0\); \(1\)
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Exercise 14.1, Question 2
Which experiments have equally likely outcomes? Explain.
(i) A driver tries to start a car: it starts or it doesn't.
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- A working car usually starts; the two outcomes are not equally likely.
Answer: Not equally likely
(ii) A player attempts a basketball shot: score or miss.
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- It depends on the player's skill.
Answer: Not equally likely
(iii) A true–false question is answered: right or wrong.
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- Guessing, each of the two answers is equally likely to be right.
Answer: Equally likely
(iv) A baby is born: boy or girl.
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- Taken as equally likely.
Answer: Equally likely
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Exercise 14.1, Question 3
Why is tossing a coin a fair way of deciding which team gets the ball first?
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- A fair coin has two equally likely outcomes, so each team has probability \(\tfrac12\), and the result can't be predicted or influenced.
Answer: Each team has an equal chance, \(\tfrac12\)
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Exercise 14.1, Question 4
Which cannot be the probability of an event? (A) \(\tfrac23\) (B) \(-1.5\) (C) 15% (D) 0.7
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- A probability lies between 0 and 1; \(-1.5\) is negative.
Answer: (B) \(-1.5\)
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Exercise 14.1, Question 5
If \(P(E) = 0.05\), what is \(P(\text{not } E)\)?
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- \(1 - 0.05\).
Answer: \(0.95\)
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Exercise 14.1, Question 6
A bag contains only lemon-flavoured sweets. One is taken without looking. Find the probability that it is:
(i) orange-flavoured
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- There are none: an impossible event.
Answer: \(0\)
(ii) lemon-flavoured
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- Every sweet is lemon: a sure event.
Answer: \(1\)
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Exercise 14.1, Question 7
The probability that 2 students in a group of 3 do not share a birthday is 0.992. What is the probability that they do?
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- Complementary event: \(1 - 0.992\).
Answer: \(0.008\)
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Exercise 14.1, Question 8
A bag has 3 red and 5 black balls. One is drawn at random. Find the probability that it is:
(i) red
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- \(\tfrac{3}{8}\).
Answer: \(\tfrac38\)
(ii) not red
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- \(1 - \tfrac38\).
Answer: \(\tfrac58\)
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Exercise 14.1, Question 9
A box has 5 red, 8 white and 4 green marbles. One is taken out at random. Find the probability that it is:
(i) red
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- 17 marbles in all.
Answer: \(\tfrac{5}{17}\)
(ii) white
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- \(\tfrac{8}{17}\).
Answer: \(\tfrac{8}{17}\)
(iii) not green
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- \(1 - \tfrac{4}{17}\).
Answer: \(\tfrac{13}{17}\)
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Exercise 14.1, Question 10
A piggy bank holds a hundred 50 p coins, fifty ₹1 coins, twenty ₹2 coins and ten ₹5 coins. One coin falls out. Find the probability that it is:
(i) a 50 p coin
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- 180 coins in all: \(\tfrac{100}{180}\).
Answer: \(\tfrac59\)
(ii) not a ₹5 coin
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- \(1 - \tfrac{10}{180}\).
Answer: \(\tfrac{17}{18}\)
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Exercise 14.1, Question 11
A tank has 5 male and 8 female fish. One is taken out at random. What is the probability it is male?
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- \(\tfrac{5}{13}\).
Answer: \(\tfrac{5}{13}\)
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Exercise 14.1, Question 12
A spinner stops at one of the numbers 1 to 8, all equally likely. Find the probability that it points at:
(i) 8
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- One of 8 outcomes.
Answer: \(\tfrac18\)
(ii) an odd number
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- 1, 3, 5, 7.
Answer: \(\tfrac12\)
(iii) a number greater than 2
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- 3 to 8: six outcomes.
Answer: \(\tfrac34\)
(iv) a number less than 9
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- All 8 outcomes: a sure event.
Answer: \(1\)
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Exercise 14.1, Question 13
A die is thrown once. Find the probability of getting:
(i) a prime number
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- 2, 3, 5.
Answer: \(\tfrac12\)
(ii) a number between 2 and 6
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- 3, 4, 5.
Answer: \(\tfrac12\)
(iii) an odd number
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- 1, 3, 5.
Answer: \(\tfrac12\)
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Exercise 14.1, Question 14
One card is drawn from a well-shuffled deck of 52. Find the probability of getting:
(i) a red king
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- King of hearts or diamonds: 2 cards.
Answer: \(\tfrac{1}{26}\)
(ii) a face card
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- J, Q, K of 4 suits: 12 cards.
Answer: \(\tfrac{3}{13}\)
(iii) a red face card
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- 6 cards.
Answer: \(\tfrac{3}{26}\)
(iv) the jack of hearts
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- 1 card.
Answer: \(\tfrac{1}{52}\)
(v) a spade
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- 13 cards.
Answer: \(\tfrac14\)
(vi) the queen of diamonds
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- 1 card.
Answer: \(\tfrac{1}{52}\)
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Exercise 14.1, Question 15
The ten, jack, queen, king and ace of diamonds are shuffled face down.
(i) One card is picked. What is the probability it is the queen?
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- 1 of 5 cards.
Answer: \(\tfrac15\)
(ii)(a) The queen is drawn and put aside. A second card is picked. Probability it is an ace?
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- 4 cards remain, one of them the ace.
Answer: \(\tfrac14\)
(ii)(b) … and probability it is a queen?
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- The queen is no longer there.
Answer: \(0\)
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Exercise 14.1, Question 16
12 defective pens are mixed with 132 good ones. One pen is taken at random. What is the probability it is good?
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- \(\dfrac{132}{144}\).
Answer: \(\tfrac{11}{12}\)
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Exercise 14.1, Question 17
(i) A lot of 20 bulbs has 4 defective ones. One is drawn at random. Probability it is defective?
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- \(\tfrac{4}{20}\).
Answer: \(\tfrac15\)
(ii) The bulb drawn in (i) is not defective and is not replaced. Another is drawn from the rest. Probability it is not defective?
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- 19 bulbs remain, 15 of them good.
Answer: \(\tfrac{15}{19}\)
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Exercise 14.1, Question 18
Discs numbered 1 to 90 are in a box. One is drawn at random. Find the probability that it bears:
(i) a two-digit number
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- 10 to 90: 81 numbers.
Answer: \(\tfrac{9}{10}\)
(ii) a perfect square
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- 1, 4, 9, …, 81: 9 numbers.
Answer: \(\tfrac{1}{10}\)
(iii) a number divisible by 5
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- 5, 10, …, 90: 18 numbers.
Answer: \(\tfrac15\)
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Exercise 14.1, Question 19
A child's die has the letters A, B, C, D, E, A on its faces. It is thrown once. Find the probability of getting:
(i) A
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- Two of the six faces show A.
Answer: \(\tfrac13\)
(ii) D
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- One face.
Answer: \(\tfrac16\)
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Exercise 14.1, Question 20
A die is dropped at random onto a 3 m × 2 m rectangle containing a circle of diameter 1 m. What is the probability it lands inside the circle?
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- Probability \[= \dfrac{\text{area of circle}}{\text{area of rectangle}} = \dfrac{\pi(0.5)^2}{6}\]
Answer: \(\dfrac{\pi}{24}\)
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Exercise 14.1, Question 21
A lot of 144 ball pens has 20 defective ones. A customer buys a pen only if it is good. One pen is taken at random. Find the probability that:
(i) she buys it
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- 124 good pens.
Answer: \(\tfrac{31}{36}\)
(ii) she does not buy it
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- \(\tfrac{20}{144}\).
Answer: \(\tfrac{5}{36}\)
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Exercise 14.1, Question 22
Two dice are thrown together and the sum is noted.
(i) Complete the table of probabilities of the sums 2 to 12.
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- 36 equally likely ordered pairs. Count pairs for each sum.
Answer: Sums 2 to 12: \[\tfrac{1}{36}, \tfrac{2}{36}, \tfrac{3}{36}, \tfrac{4}{36}, \tfrac{5}{36}, \tfrac{6}{36}, \tfrac{5}{36}, \tfrac{4}{36}, \tfrac{3}{36}, \tfrac{2}{36}, \tfrac{1}{36}\]
(ii) A student says the 11 sums are equally likely, each \(\tfrac{1}{11}\). Do you agree?
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- No: the sums are not equally likely (e.g. 7 has 6 ways, 2 has 1).
Answer: No
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Exercise 14.1, Question 23
A ₹1 coin is tossed 3 times. A player wins if all three tosses match and loses otherwise. What is the probability of losing?
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- 8 outcomes; HHH and TTT win.
- Lose \(= 1 - \tfrac28\).
Answer: \(\tfrac34\)
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Exercise 14.1, Question 24
A die is thrown twice. Find the probability that:
(i) 5 does not come up either time
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- \(5 \times 5 = 25\) of the 36 outcomes have no 5.
Answer: \(\tfrac{25}{36}\)
(ii) 5 comes up at least once
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- \(1 - \tfrac{25}{36}\).
Answer: \(\tfrac{11}{36}\)
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Exercise 14.1, Question 25
Which argument is correct? Explain.
(i) Two coins are tossed: the outcomes are two heads, two tails or one of each, so each has probability \(\tfrac13\).
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- The equally likely outcomes are HH, HT, TH, TT. 'One of each' covers two of them.
Answer: Incorrect: \(P(\text{one of each}) = \tfrac12\), the others \(\tfrac14\)
(ii) A die is thrown: the outcomes are odd or even, so \(P(\text{odd}) = \tfrac12\).
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- Three of the six equally likely faces are odd.
Answer: Correct
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