basic probability

Problem Context: Tickets in Boxes

One ticket will be drawn at random from each of the two boxes shown below:

NoteBox A

1, 2, 2, 3, 5

NoteBox B

2, 4, 4, 6

Find the probability for each of the following independent events. (Note: Probabilities are rounded to two decimal places where necessary).

Practice Questions

Part A: Both numbers are even

Find the probability that both drawn tickets have even numbers.

Correct Answer: 0.40

  1. Count the total outcomes and even numbers in each box:
    • Box A has 5 total numbers: 1, 2, 2, 3, 5. The even numbers are two twos. [P() = = 0.40]
    • Box B has 4 total numbers: 2, 4, 4, 6. All 4 numbers are even. [P() = = 1.00]
  2. Since the draws are independent, multiply their probabilities: [P() = P() P() = 0.40 = 0.40]

Part B: The sum of the numbers is exactly 6

Find the probability that the sum of the two drawn numbers equals 6.

Correct Answer: 0.25

  1. The total size of the sample space is (5 = 20).
  2. Identify all valid pairs ((A, B)) from Box A () Box B that sum to exactly 6:
    • If A = 2 (first ‘2’), B must be 4 (two choices) → 2 pairs.
    • If A = 2 (second ‘2’), B must be 4 (two choices) → 2 pairs.
    • No other elements in Box A can pair with Box B to sum to exactly 6.
  3. Calculate the probability: [P( = 6) = = 0.25]

Part C: The number from Box A is strictly greater than Box B

Find the probability that the number drawn from Box A is larger than the number drawn from Box B.

Correct Answer: 0.05

  1. Check each item in Box A against items in Box B to see where (A > B):
    • Box A elements: 1, 2, 2, 3, 5
    • Box B elements: 2, 4, 4, 6
  2. Out of 20 total combinations, only the pair ((A=5, B=2)) satisfies the requirement ((5 > 2)).
  3. Calculate probability: [P(A > B) = = 0.05]

Part D: At least one of the numbers is a 2

Find the probability that at least one of the drawn tickets shows the number 2.

Correct Answer: 0.55

Using the complement rule is the easiest way to solve “at least one” problems: [P() = 1 - P()]

  1. Find the probability of NOT drawing a 2 from each box:
    • Box A has 5 numbers, and 3 are NOT a 2 (1, 3, 5): [P() = ]
    • Box B has 4 numbers, and 3 are NOT a 2 (4, 4, 6): [P() = ]
  2. Multiply the probabilities to find the chance that neither box rolls a 2: [P() = = = 0.45]
  3. Subtract from 1 to find the remaining probability: [P() = 1 - 0.45 = 0.55]