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Probability Questions and Answers

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91.

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.

  • sum 2: 1 way
  • sum 3: 2 ways
  • sum 5: 4 ways
  • sum 7: 6 ways
  • sum 11: 2 ways

\[ 1 + 2 + 4 + 6 + 2 = 15 \quad\Longrightarrow\quad P = \frac{15}{36} = \frac{5}{12} \]

9 is not prime (\( 3 \times 3 \)) and neither is 1, which is where miscounts usually creep in.

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92.

A box has 6 black, 4 red, 2 white and 3 blue shirts. What is probability of picking at least 1 red shirt in 4 shirts that are randomly picked?

Answer: C

There are \( 6 + 4 + 2 + 3 = 15 \) shirts, of which 11 are not red. Use the complement.

\[ P(\text{no red}) = \frac{{}^{11}C_{4}}{{}^{15}C_{4}} = \frac{330}{1365} \]

\[ P(\text{at least one red}) = 1 - \frac{330}{1365} = \frac{1035}{1365} = \frac{69}{91} \]

Counting "exactly one, exactly two, exactly three, exactly four" would need four separate terms; the complement needs one.

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93.

Two friends A and B apply for a job in the same company. The chances of A getting selected is 2/5 and that of B is 4/7. What is the probability that both of them get selected?

Answer: A

The two selections are independent, so the probabilities multiply.

\[ P = \frac{2}{5} \times \frac{4}{7} = \frac{8}{35} \]

Adding them would answer a different question - and would exceed 1 in many such problems, which is the quickest sign that multiplication is meant.

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94.

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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95.

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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96.

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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97.

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.

  • Number of bedrooms - 1, 2, 3, … discrete
  • Number of bathrooms - discrete
  • Whether there is a pool - two categories, discrete
  • Sale price - any value in a range, continuous

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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98.

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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99.

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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100.

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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