What is the probability of rolling an odd number or a number less than 6 when a single die is rolled?

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To solve the problem of finding the probability of rolling an odd number or a number less than 6 with a single die, we first need to identify the outcomes of a standard die roll. A standard die has six faces, numbered from 1 to 6:

  • 1
  • 2
  • 3
  • 4
  • 5
  • 6

Next, let’s define the two events we are interested in:

Event A: Rolling an Odd Number

The odd numbers on a die are:

  • 1
  • 3
  • 5

So, Event A (rolling an odd number) can result in any of the three outcomes: 1, 3, or 5.

Event B: Rolling a Number Less Than 6

The numbers on a die that are less than 6 are:

  • 1
  • 2
  • 3
  • 4
  • 5

Thus, Event B (rolling a number less than 6) can result in five possible outcomes: 1, 2, 3, 4, or 5.

Finding the Probability of A or B

To calculate the probability of either event A or event B occurring, we can employ the principle of inclusion-exclusion:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Now, let’s compute each probability:

  • P(A): The probability of rolling an odd number:

There are 3 odd numbers (1, 3, 5) out of a total of 6 outcomes:

P(A) = 3/6 = 1/2

  • P(B): The probability of rolling a number less than 6:

There are 5 outcomes less than 6 (1, 2, 3, 4, 5):

P(B) = 5/6

  • P(A ∩ B): The probability of rolling a number that is both odd and less than 6:

The odd numbers that are also less than 6 are (1, 3, 5):

P(A ∩ B) = 3/6 = 1/2

Calculating P(A ∪ B)

We can now substitute these probabilities into our inclusion-exclusion formula:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

P(A ∪ B) = (1/2) + (5/6) – (1/2)

Calculating step by step:

  1. P(A ∪ B) = (1/2) + (5/6) – (1/2)
  2. P(A ∪ B) = (5/6)

Finally, the probability of rolling either an odd number or a number less than 6 when rolling a single die is:

P(A ∪ B) = 5/6

In conclusion, the probability of rolling an odd number or a number less than 6 is 5/6 or approximately 0.8333, which reflects a high likelihood of achieving one of these results on a single die roll.

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