Question

Mark sue and dan are trying to decide who will walk the dog and who will clean the fishbowl afterwards. they randomly pull two names out of a hat to decide where the first name is drawn will walk the dog and the second will clean the fishbowl. determine the probability the two people pulled will have the same number of letters in their first name. Assume the same person cannot be picked for both.

Answer 1

To find the probability that Mark, Sue, and Dan will randomly draw two names with the same number of letters in their first names, we first analyze the names:

- Mark has 4 letters
- Sue has 3 letters
- Dan has 3 letters

Now, we determine the total number of ways to choose two names from the three names available. The total combinations of picking 2 out of 3 is given by the binomial coefficient:

\[
\binom{3}{2} = 3
\]

The possible pairs are:
1. Mark and Sue
2. Mark and Dan
3. Sue and Dan

Next, we check which pairs have the same number of letters in their names:

- **Mark and Sue**: 4 letters (Mark) vs. 3 letters (Sue) → Not the same
- **Mark and Dan**: 4 letters (Mark) vs. 3 letters (Dan) → Not the same
- **Sue and Dan**: 3 letters (Sue) vs. 3 letters (Dan) → Same

From the above analysis, only the pair **Sue and Dan** has the same number of letters in their names.

Now, we calculate the probability that the two randomly drawn names will have the same number of letters:

\[
\text{Number of pairs with the same number of letters} = 1 \quad (\text{Sue and Dan})
\]

\[
\text{Total number of pairs} = 3
\]

Thus, the probability is:

\[
P(\text{same number of letters}) = \frac{\text{Number of pairs with the same number of letters}}{\text{Total number of pairs}} = \frac{1}{3}
\]

Therefore, the probability that the two people pulled will have the same number of letters in their first name is:

\[
\boxed{\frac{1}{3}}
\]