Expected Value: Coin flip not same as previous flip

In summary, the problem is asking for the expected number of switches in a sequence of tosses from a fair coin. This can be found by considering the expected number of Heads in the sequence, which is N/2, and then using the formula E(X) = ∑x_i*P(X=x_i). Therefore, E(X) = N/2 switches.
  • #1
DerekJ
1
0

Homework Statement


Given a sequence of Heads and Tails, let’s say that the sequence has a switch each time one toss is different than the toss before it. For instance, the sequence HHTHTTTHTH has 6 switches.

Suppose you toss a fair coin N times and record the resulting sequence of Heads and Tails. If X is the number of switches in the sequence, find E(X).

Homework Equations


We are assuming that the expected value, E(X) = ∑x_i*P(X=x_i) (the sum of each element of X, multiplied by the respective probability of that element occurring)

The Attempt at a Solution


I understand that this is going to have something to do with the number of heads and tails in the sequence and the relationship between the two. Clearly 2/n of the sequences have 0 switches, and 2/n of the sequences have n-1 switches, however I'm unsure of how to work out the values in between in general.
 
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  • #2
it's rather simple. use E(X+Y)=E(X)+E(Y)
 
  • #3
DerekJ said:

Homework Statement


Given a sequence of Heads and Tails, let’s say that the sequence has a switch each time one toss is different than the toss before it. For instance, the sequence HHTHTTTHTH has 6 switches.

Suppose you toss a fair coin N times and record the resulting sequence of Heads and Tails. If X is the number of switches in the sequence, find E(X).

Homework Equations


We are assuming that the expected value, E(X) = ∑x_i*P(X=x_i) (the sum of each element of X, multiplied by the respective probability of that element occurring)

The Attempt at a Solution


I understand that this is going to have something to do with the number of heads and tails in the sequence and the relationship between the two. Clearly 2/n of the sequences have 0 switches, and 2/n of the sequences have n-1 switches, however I'm unsure of how to work out the values in between in general.

Can you see how to do it without doing any calculations? Hint: what is the expected number of Heads in a sequence of tosses?
 

1. What is expected value in relation to a coin flip not being the same as the previous flip?

The expected value in this scenario refers to the average number of times a coin will land differently than the previous flip. In other words, it is the probability of getting a different outcome on each flip, which is 50% for a fair coin.

2. How can expected value be calculated for a coin flip not being the same as the previous flip?

To calculate the expected value, you would need to multiply the probability of a different outcome (50%) by the number of flips. For example, if you flip a coin 10 times, the expected value would be 10 x 0.5 = 5 different outcomes.

3. Is the expected value the same for all coin flips not being the same as the previous flip?

No, the expected value can vary depending on the number of flips and the probability of a different outcome. For instance, if you flip a coin 100 times, the expected value would be 50 different outcomes. However, if you flip a coin 1000 times, the expected value would be 500 different outcomes.

4. How does the concept of expected value apply to coin flips not being the same as the previous flip in real-life scenarios?

The concept of expected value can be applied to real-life scenarios where there are multiple independent events with a known probability of occurring. For example, in gambling, the expected value can help calculate the average winnings or losses over a series of bets.

5. Can the expected value of a coin flip not being the same as the previous flip be influenced by external factors?

No, the expected value is solely based on the probability of a different outcome and the number of flips, and it is not affected by external factors such as luck or previous outcomes. Each coin flip is an independent event, and the expected value remains the same regardless of external factors.

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