Probability questions that can come on GTLE

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Answers

5 parts
1

215\dfrac{2}{15}

2

12\dfrac{1}{2}

3

413\dfrac{4}{13}

4

916\dfrac{9}{16}

5
0.2

Step-by-step solution

14 steps

Part 1· 3 steps

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?

  1. Total number of balls = 4+6=104 + 6 = 10.

    Number of ways to draw 2 balls from 10 = (102)=10×92×1=45\dbinom{10}{2} = \dfrac{10 \times 9}{2 \times 1} = 45.

  2. Number of red balls = 4.

    Number of ways to draw 2 red balls from 4 = (42)=4×32×1=6\dbinom{4}{2} = \dfrac{4 \times 3}{2 \times 1} = 6.

  3. Probability both red = (42)(102)=645=215\dfrac{\dbinom{4}{2}}{\dbinom{10}{2}} = \dfrac{6}{45} = \dfrac{2}{15}.

    215\boxed{\dfrac{2}{15}}

Part 2

12\dfrac{1}{2}

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

  1. Total outcomes when rolling a die = 6.

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

  3. Probability = 36=12\dfrac{3}{6} = \dfrac{1}{2}.

    12\boxed{\dfrac{1}{2}}

Part 3

413\dfrac{4}{13}

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

  1. Total cards = 52.

  2. Kings = 4, spades = 13, king of spades counted twice, so kings or spades = 4+13−1=164 + 13 - 1 = 16.

  3. Probability = 1652=413\dfrac{16}{52} = \dfrac{4}{13}.

    413\boxed{\dfrac{4}{13}}

Part 4

916\dfrac{9}{16}

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

  1. Probability first good = 1520=34\dfrac{15}{20} = \dfrac{3}{4}.

  2. Probability second good (with replacement) = 34\dfrac{3}{4}.

  3. Probability both good = 34×34=916\dfrac{3}{4} \times \dfrac{3}{4} = \dfrac{9}{16}.

    916\boxed{\dfrac{9}{16}}

Part 50.2

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

  1. Events independent.

  2. P(both)=P(Monday)×P(Tuesday)=0.4×0.5=0.2P(\text{both}) = P(\text{Monday}) \times P(\text{Tuesday}) = 0.4 \times 0.5 = 0.2.

    0.2\boxed{0.2}
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