Probability questions that can come on GTLE

Question 1: A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?

Step 1: Total number of balls = $4 + 6 = 10$.

Number of ways to draw 2 balls from 10 = $\dbinom{10}{2} = \dfrac{10 \times 9}{2 \times 1} = 45$.

Step 2: Number of red balls = 4.

Number of ways to draw 2 red balls from 4 = $\dbinom{4}{2} = \dfrac{4 \times 3}{2 \times 1} = 6$.

Step 3: Probability both red = $\dfrac{\dbinom{4}{2}}{\dbinom{10}{2}} = \dfrac{6}{45} = \dfrac{2}{15}$.

$\boxed{\dfrac{2}{15}}$

Question 2: A fair die is rolled. What is the probability of getting an even number?

Step 1: Total outcomes when rolling a die = 6.

Step 2: Even numbers = 2, 4, 6, so 3 favorable outcomes.

Step 3: Probability = $\dfrac{3}{6} = \dfrac{1}{2}$.

$\boxed{\dfrac{1}{2}}$

Question 3: From a standard deck of 52 cards, one card is drawn at random. What is the probability it is a king or a spade?

Step 1: Total cards = 52.

Step 2: Kings = 4, spades = 13, king of spades counted twice, so kings or spades = $4 + 13 - 1 = 16$.

Step 3: Probability = $\dfrac{16}{52} = \dfrac{4}{13}$.

$\boxed{\dfrac{4}{13}}$

Question 4: A box contains 5 defective and 15 good items. Two items are drawn with replacement. Find the probability both are good.

Step 1: Probability first good = $\dfrac{15}{20} = \dfrac{3}{4}$.

Step 2: Probability second good (with replacement) = $\dfrac{3}{4}$.

Step 3: Probability both good = $\dfrac{3}{4} \times \dfrac{3}{4} = \dfrac{9}{16}$.

$\boxed{\dfrac{9}{16}}$

Question 5: The probability of rain on Monday is 0.4, Tuesday 0.5, independently. Find probability of rain on both days.

Step 1: Events independent.

Step 2: $P(both) = P(Monday) \times P(Tuesday) = 0.4 \times 0.5 = 0.2$.

$\boxed{0.2}$