An ant starts a random walk on the real number line at 0. At each step, the ant moves by +1 or −1 with equal probability. After 6 moves, the probability that the ant is on a positive number can be expressed as a/b, where a and b are positive coprime integers. What is the value of a+b?

Well, you know what they say about ants and their random walks on the number line, right? They really know how to make things interesting!

Now, let's tackle this probability puzzle. After 6 moves, we want to find the probability that the ant ends up on a positive number.

Let's consider the possibilities. Out of the 6 moves, for the ant to end up on a positive number, it must have taken more steps towards the positive side than towards the negative side.

So, in order to calculate this probability, we need to count the number of ways the ant can take more steps towards the positive side than the negative side.

For the first step, the ant can either move +1 or -1, no restrictions there. For the second step, it can also move +1 or -1, same story. It can either move in the same direction as the first step or change its direction.

Now, when we get to the third step, things start to get interesting. We need to keep track of the number of steps towards the positive and negative sides.

Let's break it down step by step:

Step 1: +1 or -1 (equally likely)
Step 2: +1 or -1 (equally likely)
Step 3: +1, -1, or we can have the same step as step 1 (2 ways)

Now, in order to have more steps towards the positive side, the ant needs to have taken more +1 steps than -1 steps so far. This means that out of the first 3 steps, we need an odd number of +1 steps.

Out of the 2 possibilities (either +1 or -1) for each of the first 2 steps, there are 2 combinations that give us an odd number of +1 steps: (++ or --).

Now, let's move on to the next 3 steps:

Step 1: +1 or -1 (equally likely)
Step 2: +1 or -1 (equally likely)
Step 3: +1 (Fixed because the previous 3 steps already gave us an odd number of +1 steps)

Again, if we want more steps towards the positive side, we need to have an odd number of +1 steps out of the first 3 steps. Since step 3 is fixed, we only have two possibilities left: ++ or --.

So, out of the 4 remaining steps, 2 of them must be +1.

That means, out of the total 6 steps, we have 4 +1's and 2 -1's.

Now, let's calculate the probability!

The number of ways to arrange 4 +1's and 2 -1's is given by the binomial coefficient (6 choose 4), which is equal to 15.

The total number of possible outcomes is 2^6 (since the ant can either move +1 or -1 at each step), which is equal to 64.

So, the probability that the ant is on a positive number after 6 moves is 15/64.

Therefore, a = 15 and b = 64.

And finally, if we add a and b together, we get 15 + 64 = 79.

So, the value of a+b is 79.

Hope this clarifies things for you!

To find the probability that the ant is on a positive number after 6 moves, we can use the concept of binomial distribution.

Let's consider the ant's position after 6 moves. Since the ant can move either +1 or -1 with equal probability at each step, the number of times it moves in a positive direction can vary from 0 to 6.

To find the probability of being on a positive number after 6 moves, we need to sum up the probabilities of each possible number of positive movements.

When the ant moves 0 times in the positive direction, it must have moved 6 times in the negative direction. The number of ways this can happen is given by the binomial coefficient, which is denoted as C(n, k), representing choosing k items out of n. In this case, we have C(6, 0) = 1.

When the ant moves 1 time in the positive direction, it must have moved 5 times in the negative direction. The number of ways this can happen is C(6, 1) = 6.

Similarly, for 2 positive movements, we have C(6, 2) = 15.

For 3 positive movements, C(6, 3) = 20.

For 4 positive movements, C(6, 4) = 15.

For 5 positive movements, C(6, 5) = 6.

And for 6 positive movements, C(6, 6) = 1.

Therefore, the total number of ways the ant can end up on a positive number is 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64.

Since each step has an equal probability of being +1 or -1, the total number of possible outcomes after 6 steps is 2^6 = 64.

Hence, the probability of the ant being on a positive number after 6 moves is 64/64 = 1.

So, the value of a + b is 1 + 1 = 2.

To solve this problem, we need to calculate the probability that the ant is on a positive number after 6 moves.

The ant can either move +1 or -1 on each step, and after 6 moves, there are 2^6 = 64 possible paths the ant can take. We can represent these paths using a binary sequence, where 0 represents a step of -1 and 1 represents a step of +1.

For the ant to end up on a positive number after 6 moves, it needs to have taken more steps of +1 than -1. Since there are 6 moves in total, at least 4 of them need to be +1's.

We can count all the possible binary sequences with at least 4 ones:

- 4 ones and 2 zeros: There are 6 choose 4 ways to arrange 4 ones and 2 zeros, which is given by the formula 6! / (4! * 2!) = 15.

- 5 ones and 1 zero: There are 6 choose 5 ways to arrange 5 ones and 1 zero, which is given by the formula 6! / (5! * 1!) = 6.

- 6 ones and 0 zeros: There is only 1 way to arrange 6 ones.

So, the total number of paths where the ant ends up on a positive number is 15 + 6 + 1 = 22.

Since each path has an equal probability of occurring (1/64), the probability that the ant is on a positive number after 6 moves is 22/64 = 11/32.

Therefore, the value of a+b is 11+32 = 43.

3/8 so 3+8=11