25 independent, fair coins are tossed in a row. What is the expected number of consecutive HH pairs?
Details and assumptions
If 6 coin tosses in a row give HHTHHH, the number of consecutive HH pairs is 3.
12
how
To find the expected number of consecutive HH pairs, we'll need to consider the possible outcomes of the coin tosses.
Let's break down the problem by thinking about each coin toss individually. In any given toss, there are two possible outcomes: heads (H) or tails (T). Since each coin toss is independent and fair, the probability of getting heads or tails is each 0.5.
Now, let's consider the first coin toss. It can either result in heads (H) or tails (T). In this case, we're interested in the possibility of getting a consecutive HH pair. Since it's the first toss, we can't have a consecutive pair yet. Therefore, the number of consecutive HH pairs is zero.
Moving on to the second coin toss, we have two possible outcomes: HH or TH. If we get HH, we have one consecutive HH pair. If we get TH, we have zero consecutive HH pairs. Since both outcomes have equal probabilities, the expected number of consecutive HH pairs after the second coin toss is 0.5.
Similarly, we can extend this reasoning for the remaining coin tosses. Let's consider the cases for the consecutive HH pairs after the third toss:
1. HHH: We have one consecutive HH pair.
2. HTH: We have zero consecutive HH pairs.
Again, both outcomes have equal probabilities, so the expected number of consecutive HH pairs after the third coin toss is also 0.5.
By following this logic, we see that the expected number of consecutive HH pairs after each coin toss is 0.5. Since we have 25 independent coin tosses, the total expected number of consecutive HH pairs is:
0.5 (expected number of consecutive HH pairs per toss) × 25 (number of tosses) = 12.5
Therefore, the expected number of consecutive HH pairs when tossing 25 independent, fair coins in a row is 12.5.