en committee members each will vote either A or B to become their leader.
Suppose A receives 4 votes and B receives 6 votes. Ten votes are randomly
ordered.
(a) What is the probability that A and B have an equal number of votes
after 6 votes are revealed?

To solve this problem, we need to determine the probability that A and B have an equal number of votes after 6 votes are revealed out of the 10 votes in total. Given that A received 4 votes and B received 6 votes, the key is to find the number of ways in which the votes can be ordered so that A and B each have 3 votes among the first 6 revealed votes.

### Step 1: Define the Problem Mathematically
- Total votes: 10 (4 for A, 6 for B)
- We want to determine the probability that, after 6 votes are revealed, A has 3 votes and B has 3 votes.

### Step 2: Count the Total Combinations of the First 6 Votes
We first choose 6 votes out of the total 10 votes. The number of ways to do this is:

\[
\binom{10}{6}
\]

### Step 3: Count the Favorable Combinations
Now, within these 6 votes, we need exactly 3 votes for A and 3 votes for B. The number of ways to arrange 3 votes for A out of A's 4 total votes is:

\[
\binom{4}{3} = 4
\]

Similarly, the number of ways to arrange 3 votes for B out of B's 6 total votes is:

\[
\binom{6}{3} = 20
\]

Therefore, the total number of favorable outcomes where A and B both have 3 votes among the first 6 votes is:

\[
\binom{4}{3} \times \binom{6}{3} = 4 \times 20 = 80
\]

### Step 4: Calculate the Probability
The probability is the ratio of the number of favorable outcomes to the total number of outcomes:

\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{80}{\binom{10}{6}}
\]

Now, calculate \(\binom{10}{6}\):

\[
\binom{10}{6} = \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

So, the probability is:

\[
P = \frac{80}{210} = \frac{8}{21}
\]

### Final Answer
The probability that A and B have an equal number of votes after 6 votes are revealed is:

\[
\boxed{\frac{8}{21}}
\]

Would you like any further details or have any questions?

### Related Questions:
1. What is the probability that A leads B after 6 votes?
2. How would the probability change if A received 5 votes and B received 5 votes?
3. What is the probability that B wins overall given the 4-6 split?
4. What is the probability of A receiving exactly 2 votes in the first 6 votes?
5. How can this problem be extended if there are more than two candidates?

**Tip:** When calculating probabilities with binomial coefficients, break the problem into steps to count the total and favorable outcomes separately.

Ten committee members each will vote either A or B to become their leader. Suppose A receives 4 votes and B receives 6 votes. Ten votes are randomly ordered. (a) What is the probability that A and B have an equal number of votes after 6 votes are revealed?

Discover the probability that A and B have an equal number of votes after 6 votes are revealed in a scenario where A has 4 total votes and B has 6 total votes. This problem uses combinatorics and binomial coefficients to solve for the probability, suitable for students in Grades 10-12.

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