The probability that it will rain tomorrow is 0.85. What is the probability that it will not rain tomorrow
Answer: C
An event and its complement together account for every possibility, so their probabilities add to 1.
\[ P(\text{not rain}) = 1 - P(\text{rain}) = 1 - 0.85 = 0.15 \]
\[ 0.15 = \frac{15}{100} = \frac{3}{20} \]
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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?
Answer: A
The bag holds \( 2 + 3 + 2 = 7 \) balls, of which 5 are not blue.
\[ P = \frac{{}^{5}C_{2}}{{}^{7}C_{2}} = \frac{10}{21} \]
Counting the non-blue balls directly is quicker than working through the red and green cases separately.
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24 cards numbered 1, 2, 3, …., 23, 24 are put in a box and mixed thoroughly. One person draws a card from the box. The probability that the number on the card is divisible by 2 or 3 or both is
Answer: B
The total possible outcomes = 24
Numbers divisible by only 2 are 2, 4, 8, 10, 14, 16, 20, 22 (8 numbers) --- (1)
Numbers divisible only by 3 are 3, 9, 15, 21 (4 numbers) --- (2)
Numbers divisible by both 2 and 3 are 6, 12, 18, 24 (4 numbers) --- (3)
From (1), (2) and (3), we see that the number of favourable outcomes is 16 (i.e., 8 + 4 + 4).
We know that, Probability of an event E, P (E) =number of favourable outcomes/ total number of outcomes
=16/24
=2/3
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What is the probability of not picking a face card when you draw a card at random from a pack of 52 cards?
Answer: C
There are 12 face cards - jack, queen and king in each of the four suits - so 40 cards are not face cards.
\[ P = \frac{40}{52} = \frac{10}{13} \]
Equivalently \( 1 - \tfrac{12}{52} = 1 - \tfrac{3}{13} = \tfrac{10}{13} \).
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A dice is rolled twice. What is the probability of getting sum 9?
Answer: C
Two rolls give 36 outcomes. Those totalling 9 are
\[ (3,6),\ (4,5),\ (5,4),\ (6,3) \]
\[ P = \frac{4}{36} = \frac{1}{9} \]
The order of the two rolls matters, so \( (3,6) \) and \( (6,3) \) are separate outcomes; treating them as one would halve the count.
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Which of the following is not a correct statement about a probability
Answer: D
A probability is a number attached to an event, not a collection of experiments.
The three true statements are that it lies between 0 and 1, that it may be written as a decimal or a fraction, and that a value near 0 marks an unlikely event.
The false one is that it is the collection of several experiments - that describes a sequence of trials, not the probability itself.
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An event is very unlikely to happen. Its probability is closest to
Answer: A
Probabilities run from 0 (impossible) to 1 (certain), so the less likely an event, the closer its probability sits to 0.
\[ 0.0001 < 0.001 < 0.01 < 0.1 \]
The smallest value, 0.0001, describes the least likely event - about one chance in ten thousand.
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Two dice are tossed. The probability that the total score is a prime number is:
Answer: B
Two dice give 36 outcomes. The prime totals are 2, 3, 5, 7 and 11:
\[ 1 + 2 + 4 + 6 + 2 = 15 \text{ outcomes} \]
\[ P = \frac{15}{36} = \frac{5}{12} \]
9 is not prime, and neither is 1 - the two totals most often included by mistake.
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What is the probability that the minute and hour hands of a clock will form an acute angle at any given time?
Answer: C
The amount of time in a period of 12 hours when the hands will form an acute angle will be the same as that for obtuse angle. So, you would think that the probability is 12. But if you take into account the small amounts of time when the hands are aligned ( 0∘), hands are at right angles and hands are facing in opposite directions (180∘), then the probability would be slightly less than 0.5.
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Two dice are tossed simultaneously. Find the probability that the total is a prime number.
Answer: B
Two dice give 36 outcomes. The prime totals are 2, 3, 5, 7 and 11.
\[ 1 + 2 + 4 + 6 + 2 = 15 \quad\Longrightarrow\quad P = \frac{15}{36} = \frac{5}{12} \]
9 is not prime (\( 3 \times 3 \)) and neither is 1, which is where miscounts usually creep in.
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