If a card is chosen from a standard deck of cards, what is the probability of getting a five or a seven?
Answer: C
A standard deck has four fives and four sevens, and no card is both.
\[ P = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13} \]
Because the two events are mutually exclusive, the probabilities simply add with no overlap to subtract.
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The probability that a prime number selected at random from the numbers (1,2,3, ..........35) is :
Answer: B
The primes up to 35 are
\[ 2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29,\ 31 \]
which is 11 numbers out of 35.
\[ P = \frac{11}{35} \]
1 is not prime - counting it would give 12 and the first distractor.
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An experiment whose outcomes has to be among a set of events that are completely known but whose exact outcomes is unknown is a
Answer: C
A random experiment is one whose set of possible outcomes is known in advance but whose particular outcome on any trial cannot be predicted - rolling a die, for instance.
The other terms name different things: the sample space is the set of outcomes, and an elementary event is a single one of them. Neither is the experiment itself.
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Which of the following is not an example of a discrete probability distribution?
Answer: A
A discrete distribution takes values that can be listed - whole counts or categories.
So the sale or purchase price of a house is the odd one out: between any two prices there is always another possible price.
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What is the probability that a number selected from the numbers (1, 2, 3,..........,15) is a multiple of 4?
Answer: A
The multiples of 4 between 1 and 15 are 4, 8 and 12 - three of them.
\[ P = \frac{3}{15} = \frac{1}{5} \]
16 is the next multiple but lies outside the range, which is what keeps the count at three.
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A number is chosen at random among the first 100 natural numbers. Find the probability that the number chosen is prime.
Answer: A
There are 25 primes among the first 100 natural numbers.
\[ P = \frac{25}{100} = \frac{1}{4} \]
The count of 25 is worth remembering: there are 4 primes below 10, 21 more up to 100, and 1 is not among them.
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The probability that a number selected at random from the first 50 natural numbers is a composite number is -.
Answer: B
Split the first 50 natural numbers into three groups: the number 1, the primes, and the composites.
\[ \text{primes up to 50} = 15 \]
\[ \text{composites} = 50 - 15 - 1 = 34 \]
\[ P = \frac{34}{50} = \frac{17}{25} \]
1 is neither prime nor composite, so it must be excluded separately - forgetting it gives 35 and \( \tfrac{7}{10} \).
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A card is drawn from a well-shuffled deck of playing cards. Find the probability of drawing a black card which is neither a face card nor an ace?
Answer: C
In each suit, there are 9 cards that are not face cards and ace.
Hence, there will be a total of 18 cards in a deck which are black and are not face cards and ace.
We know that, Probability of an event E, P (E) =number of favourable outcomes/total number of outcomes
Required probability is 18/52 = 9/26.
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In a drawer there are 4 white socks, 3 blue socks and 5 grey socks. Two socks are picked randomly. What is the possibility that both the socks are of same color?
Answer: D
There are \( 4 + 3 + 5 = 12 \) socks in all.
\[ \text{total ways} = {}^{12}C_{2} = 66 \]
Same colour, counted by colour:
\[ {}^{4}C_{2} + {}^{3}C_{2} + {}^{5}C_{2} = 6 + 3 + 10 = 19 \]
\[ P = \frac{19}{66} \]
The three colours are mutually exclusive, so their counts simply add.
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A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?
Answer: B
The bag holds \( 6 + 8 = 14 \) balls.
\[ P = \frac{8}{14} = \frac{4}{7} \]
The denominator is the total, not the number of black balls - using 6 would give \( \tfrac{4}{3} \), which is not even a valid probability.
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