Daily Math Problem

1Easyprobability

Mr. Chen has a jar with $5$ red marbles and $3$ blue marbles. He randomly draws $2$ marbles without replacement. What is the probability that both marbles are the same color?

Express as a fraction in lowest terms

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Count same-color pairs:

\binom{5}{2} + \binom{3}{2} = 10 + 3 = 13

. Total pairs:

\binom{8}{2} = 28

. Probability

= \frac{13}{28}

.

2Mediumprobability

Using the same jar ( $5$ red, $3$ blue marbles), Mr. Chen draws marbles one at a time without replacement until he gets a red marble. What is the expected number of draws?

Express as a fraction or decimal

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By symmetry, the expected position of the first red marble among 8 total is

\frac{8+1}{5+1} = \frac{9}{6} = \frac{3}{2}

.

3Hardprobability

Mr. Chen has a jar with $5$ red and $3$ blue marbles. He draws without replacement until he has drawn at least $2$ red marbles. What is the expected number of draws?

Enter a whole number

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Using the negative hypergeometric distribution:

E = \frac{(N+1) \cdot r}{K+1} = \frac{9 \cdot 2}{6} = 3

.

4Hardprobability

Mr. Chen adds $k$ green marbles to a jar already containing $5$ red and $3$ blue marbles. He draws $2$ marbles without replacement. Find the value of $k$ such that the probability both draws are the same color equals $\frac{2}{9}$ .

Enter a positive integer

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Set up

P(\text{same color}) = \frac{\binom{5}{2}+\binom{3}{2}+\binom{k}{2}}{\binom{8+k}{2}} = \frac{2}{9}

. Numerator:

13 + \frac{k(k-1)}{2}

. Denominator:

\frac{(8+k)(7+k)}{2}

. Solving gives

k=4

.

Mr. Chen's Marble Jar