A fair die is rolled once. What is the probability of getting an even number?

12. Even outcomes are 2, 4 and 6 — three of the six equally likely faces, so the probability is 36 = 12.

Two fair coins are tossed together. What is the probability of getting exactly one head?

12. The equally likely outcomes are HH, HT, TH, TT. Exactly one head occurs in HT and TH, giving 24 = 12.

A card is drawn from a well-shuffled pack of 52. What is the probability that it is a face card (Jack, Queen or King)?

313. There are 3 4 = 12 face cards, so the probability is 1252 = 313.

Two dice are thrown together. What is the probability that the sum of the numbers is 8?

536. The pairs summing to 8 are (2,6), (3,5), (4,4), (5,3) and (6,2) — 5 of the 36 equally likely outcomes, so the probability is 536.

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Probability — Questions & Answers

For equally likely outcomes, probability is favourable outcomes over total outcomes. Combinations count the outcomes for 'draw $k$ from $n$ ' questions; the complement often shortens 'at least one' problems.

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Key formulas

$P(E) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ , with $0 \le P(E) \le 1$
$P(\text{not } E) = 1 - P(E)$
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$ ; for mutually exclusive events $P(A \cap B) = 0$

5 solved Probability questions

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01
A fair die is rolled once. What is the probability of getting an even number?
5 steps · reveal one at a time
1. Step 1: Understand Probability
  Probability is calculated by dividing the number of favorable outcomes (what we want) by the total number of possible outcomes. Each outcome must be equally likely for this formula to apply.
2. Step 2: Total Outcomes
  When a fair die is rolled, there are six possible outcomes. These are the numbers $1, 2, 3, 4, 5,$ and $6$ . So, the total number of possible outcomes is $6$ .
3. Step 3: Favorable Outcomes
  We are looking for the probability of getting an even number. On a standard die, the even numbers are $2, 4,$ and $6$ . Thus, there are $3$ favorable outcomes.
4. Step 4: Calculate Probability
  Now, we can calculate the probability. Divide the number of favorable outcomes ( $3$ ) by the total number of possible outcomes ( $6$ ). This gives us $\frac{3}{6}$ , which simplifies to $\frac{1}{2}$ .
5. Step 5: The answer
  So the answer is (C) $\dfrac{1}{2}$ . This represents a 50% chance of rolling an even number on a fair die.
02
Two fair coins are tossed together. What is the probability of getting exactly one head?
5 steps · reveal one at a time
1. Step 1: List Outcomes
  When two fair coins are tossed, each coin can land on Heads (H) or Tails (T). Let's list all the possible combinations for the two coins.
2. Step 2: Total Outcomes
  The possible outcomes are HH (Head, Head), HT (Head, Tail), TH (Tail, Head), and TT (Tail, Tail). There are $4$ equally likely outcomes in total.
3. Step 3: Favorable Outcomes
  We need to find the probability of getting exactly one head. Looking at our list, the outcomes with exactly one head are HT and TH. There are $2$ favorable outcomes.
4. Step 4: Calculate Probability
  The probability is the number of favorable outcomes divided by the total number of outcomes. So, the probability is $\frac{2}{4}$ , which simplifies to $\frac{1}{2}$ .
5. Step 5: The answer
  So the answer is (B) $\frac{1}{2}$ . This is because there are 2 ways to get exactly one head (HT, TH) out of 4 total possible outcomes (HH, HT, TH, TT).
03
A card is drawn from a well-shuffled pack of 52. What is the probability that it is a face card (Jack, Queen or King)?
5 steps · reveal one at a time
1. Step 1: Total Cards
  A standard deck of cards contains 52 cards in total. This is the total number of possible outcomes when drawing a single card.
2. Step 2: Identify Face Cards
  Face cards in a standard deck are the Jack, Queen, and King. There are four suits: hearts, diamonds, clubs, and spades.
3. Step 3: Count Favorable Cards
  For each of the three face card types (Jack, Queen, King), there is one card in each of the four suits. So, the total number of face cards is $3 \times 4 = 12$ .
4. Step 4: Calculate Probability
  The probability is the number of favorable outcomes (face cards) divided by the total number of cards. This gives us $\frac{12}{52}$ .
5. Step 5: The answer
  So the answer is (D) $\dfrac{3}{13}$ . We simplify the fraction $\frac{12}{52}$ by dividing both the numerator and denominator by 4, which results in $\frac{3}{13}$ .
04
A bag contains 4 red and 6 blue balls. Two balls are drawn at random. What is the probability that both are red?
5 steps · reveal one at a time
1. Step 1: Identify total balls
  First, let's note the total number of balls in the bag. There are 4 red balls and 6 blue balls, making a total of 10 balls. We need to find the probability of drawing two red balls.
2. Step 2: Total ways to draw
  We are drawing 2 balls from a total of 10. The number of ways to choose 2 balls from 10 (where order doesn't matter) is given by the combination formula $^nC_r = \frac{n!}{r!(n-r)!}$ . So, the total number of ways is $^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45$ .
3. Step 3: Favorable ways to draw
  Next, we need to find the number of ways to draw 2 red balls. There are 4 red balls in total. The number of ways to choose 2 red balls from these 4 is $^4C_2 = \frac{4 \times 3}{2 \times 1} = 6$ .
4. Step 4: Calculate probability
  The probability of drawing two red balls is the ratio of the number of favorable ways (drawing 2 red balls) to the total number of possible ways (drawing any 2 balls). This is $\frac{^4C_2}{^{10}C_2} = \frac{6}{45}$ .
5. Step 5: The answer
  So the answer is (A) $\dfrac{2}{15}$ . When we simplify the fraction $\frac{6}{45}$ by dividing both the numerator and the denominator by their greatest common divisor, 3, we get $\frac{6 \div 3}{45 \div 3} = \frac{2}{15}$ .
05
Two dice are thrown together. What is the probability that the sum of the numbers is 8?
4 steps · reveal one at a time
1. Step 1: Find Total Outcomes
  When two dice are thrown, each die can land on 1, 2, 3, 4, 5, or 6. To find the total number of possible outcomes, we multiply the number of outcomes for each die: $6 \times 6 = 36$ .
2. Step 2: List Favorable Outcomes
  Now, let's list all the pairs of numbers from the two dice that add up to 8. These are: (2,6), (3,5), (4,4), (5,3), and (6,2). There are 5 such favorable outcomes.
3. Step 3: Calculate Probability
  The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. So, the probability is $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{36}$ .
4. Step 4: The answer
  So the answer is (C) $\dfrac{5}{36}$ , because there are 5 pairs that sum to 8 out of 36 total possible outcomes when two dice are thrown.

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