Question

There are five red chips and three blue chips in a bowl. The red chips are numbered \(1,2,3,4,5\), respectively, and the blue chips are numbered \(1,2,3\), respectively. If two chips are to be drawn at random and without replacement, find the probability that these chips have either the same number or the same color,

Short answer

The probability that the chips drawn have either the same number or the same color is \(4 / 7\).

Step-by-step solution

Calculate the total possible outcomes

When two chips are drawn without replacement from a bowl containing 8 chips, the total number of ways this can be done is \({{8}\choose{2}}\), or 8 choose 2. Using the combination formula for calculating number of ways to choose \(r\) objects from \(n\), we get total number of outcomes as 28.

Determine the possible outcomes where chips have the same color

Same color means either both are red or both are blue. There are 5 red chips and 3 blue chips. The number of ways to choose 2 red chips from 5 is \({{5}\choose{2}}\), or 5 choose 2, which equals to 10. The number of ways to choose 2 blue from 3 is \({{3}\choose{2}}\), or 3 choose 2, which equals to 3. So, total number of outcomes where both chips have same colors equals to 10 + 3 = 13.

Determine the possible outcomes where chips have the same number

The chips can have same numbers only when one is red and the other is blue. And since there are 3 chips numbered 1, 2, 3 of each color respectively, there are total 3 such outcomes.

Find the probability

The number of favorable outcomes are the outcomes where both chips either have same number or same color, which equals to 13 (same color) + 3 (same number) = 16. Probability is calculated as number of favorable outcomes divided by total number of outcomes. Hence, the desired probability = 16 / 28 = 4 / 7.