25 independent, fair coins are tossed in a row. What is the expected number of consecutive HH pairs?
If 6 coin tosses in a row give HHTHHH, the number of consecutive HH pairs is 3.
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To find the expected number of consecutive HH pairs in 25 coin tosses, we can use the concept of linearity of expectation. Let's break down the problem into smaller parts.
Consider a single coin toss. There are two possible outcomes: heads (H) or tails (T). Since we are interested in consecutive HH pairs, we need to focus on the H outcomes.
The probability of getting an H outcome on a single coin toss is 1/2. So, the probability of getting an HH pair on two consecutive coin tosses is (1/2) * (1/2) = 1/4.
Now, let's consider two consecutive coin tosses. The probability of getting an HH pair on these two tosses is 1/4, as calculated above.
If we continue this process, we can determine the probabilities of getting HH pairs on larger sequences of coin tosses. Specifically, for any k consecutive tosses, the probability of getting an HH pair is (1/4)^k.
Given that there are 25 independent coin tosses, we need to consider sequences of 2, 3, 4, and so on, up to 25 consecutive tosses. We can calculate the expected number of HH pairs by multiplying each probability of getting an HH pair on k consecutive tosses by the number of possible sequences of k consecutive tosses.
To calculate the number of sequences for a given k, we subtract k from the total number of tosses (25) and add 1. For example, for k = 2, there are 25 - 2 + 1 = 24 possible sequences.
Using this approach, we can calculate the expected number of HH pairs as follows:
Expected number = (probability of HH pair on 2 consecutive tosses * number of sequences for k = 2)
+ (probability of HH pair on 3 consecutive tosses * number of sequences for k = 3)
+ ...
+ (probability of HH pair on 25 consecutive tosses * number of sequences for k = 25)
Expected number = (1/4)^2 * (25-2+1) + (1/4)^3 * (25-3+1) + ... + (1/4)^25
Calculating this expression will give you the expected number of consecutive HH pairs in 25 independent, fair coin tosses.