If two ratios are formed at random from the four numbers 1, 2, 4, and 8, what is the probability that the ratios are equal?

Please show your work.

The question asks about the probability that two ratios formed at random from the numbers 1, 2, 4, and 8 are equal. To find the answer, we need to consider how many unique ratios can be formed with these numbers and how many of these are equal to one another.
Step-by-step Explanation

First, we list down all possible ratios using the four numbers in pairs: 1/2, 1/4, 1/8, 2/1, 2/4, 2/8, 4/1, 4/2, 4/8, 8/1, 8/2, 8/4. These are the ratios before simplifying them.
After simplifying, we'll find that some ratios will be the same: 1/2 is the same as 4/8, 2/1 is the same as 8/4, and so on.
We find there are 6 different simplified ratios: 1/2, 1/4, 1/8, 2/1, 4/1, and 8/1.
Next, we calculate the number of unique pairs of ratios that can be formed which are (6 * 6)/2 = 18, since order does not matter.
We then count the number of equal ratio pairs. There are 6 pairs where the ratios are equal (because each ratio can be paired with itself).
Finally, we calculate the probability that the ratios are equal by dividing the number of equal pairs by the total number of pairs, which gives 6/18 = 1/3.

To round this to four decimal places, the probability is 0.3333.

Answered by qwarrow | 2024-06-19

The probability that two randomly formed ratios from the numbers 1, 2, 4, and 8 are equal is 3 1 ​ , or approximately 0.3333. This is calculated by counting the total pairs of unique ratios and the number of equal pairs. Ultimately, we find that there are 7 equal pairs out of a total of 21 possible pairs.
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Answered by qwarrow | 2024-12-23