Coins numbered 1-200 are placed in a hat. What is the probability that a coin chosen at random out of the hat is a multiple of 6 or 9?

multiples of 6:

6 12 18 .... 198 ---> 33 of them

multiples of 9
9 18 27 ... 198 ----> 22 of them

multiples of 18 , their LCM
18 36 ..198 ----> 11 of them

number of multiple of 6 OR 9 = 33 + 22 - 11 = 44

prob(multiple of 6 OR 9) = 44/200 = 11/50 or .22

To find the probability that a randomly chosen coin from the hat is a multiple of 6 or 9, we need to determine the total number of favorable outcomes and divide it by the total number of equally likely outcomes.

First, let's count the total number of coins in the hat, which is 200.

Next, we need to find the number of coins that are multiples of 6 or 9. We can do this by dividing the range [1, 200] into smaller ranges using the least common multiple (LCM) of 6 and 9, which is 18.

Every 18th coin is a multiple of 6, and every 18th coin is also a multiple of 9. However, some coins will be counted twice since they are multiples of both 6 and 9. To avoid counting these coins twice, we need to subtract the number of coins which are multiples of 6 and 9 simultaneously, which is the same as finding the least common multiple of 6 and 9.

The multiples of 6 are: 6, 12, 18, 24, ..., 198. The multiples of 9 are: 9, 18, 27, ..., 198.

Notice that for every 18th multiple of 6, only every 9th multiple of 6 is also a multiple of 9. Therefore, we need to count only every 18th multiple of 6 to avoid duplicates.

We can calculate the probability as follows:

Number of favorable outcomes = (Number of multiples of 6) + (Number of multiples of 9) - (Number of multiples of both 6 and 9)

Number of multiples of 6 = 200/6 = 33.33 (rounded to 33)

Number of multiples of 9 = 200/9 = 22.22 (rounded to 22)

Number of multiples of both 6 and 9 = 200/18 = 11.11 (rounded to 11)

Number of favorable outcomes = 33 + 22 - 11 = 44

Total number of equally likely outcomes = 200

Therefore, the probability that a coin chosen at random out of the hat is a multiple of 6 or 9 is:

Probability = Number of favorable outcomes / Total number of equally likely outcomes

Probability = 44/200

Simplifying the fraction, we get:

Probability = 11/50

Hence, the probability is 11/50.