Question

A box contains 6 red marbles numbered 1 through 6 and 4 white marbles numbered from 12 through 15. Find the probability that a marble drawn is red or even numbered.

Answer 1

Total number of marbles = (6 + 4) = 10
Let S be the sample space.
Then n(S) = number of ways of selecting one marble out of 10 = ¹⁰C₁ = 10 ways

Let E₄ = event of getting a red marble
i.e. E₄ = {1, 2, 3, 4, 5, 6}
∴ n(E₄) = 6
Now, P(E₄) =\[\frac{6}{10} = \frac{3}{5}\]    ................(i) 

Let E₅ = event of getting even numbered marble
Then E₅ = {2, 4, 6, 12, 14}
i.e.n(E₅) = 5
Now, P(E₅) = \[\frac{5}{10} = \frac{1}{2}\]

From (i) and (ii), we get:
E₄ ∩ E₅ = {2, 4, 6}

\[\Rightarrow\] n(E₄ ∩ E₅) = 3

 

\[\Rightarrow\]  P(E₄ ∩ E₅) = \[\frac{3}{10}\]

By addition theorem, we have:
P (E₄ ∪ E₅) = P(E₄) + P (E₅) − P (E₄ ∩ E₅)
⇒ P (E₄ ∪ E₅) =

\[\frac{3}{5} + \frac{1}{2} - \frac{3}{10} = \frac{8}{10} = \frac{4}{5}\] 

Hence, required probability = P(E₄ ∪ E₅) = \[\frac{4}{5}\]