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Chapter 7: Problem 2

What is the expected number of heads that come up when a fair coin is flipped 10 times?

Short Answer

Expert verified
The expected number of heads is 5.

Step by step solution

01

Understand the Problem

A fair coin has an equal probability of landing heads or tails, which is 0.5 for each flip. We need to calculate the expected number of heads when the coin is flipped 10 times.
02

Define Variables

Let the random variable for one coin flip be denoted as X. X can be 1 if the result is heads or 0 if the result is tails.
03

Calculate the Expected Value

The expected value (mean) for one flip of the coin is given by \[ E(X) = 0.5(1) + 0.5(0) = 0.5 \].
04

Determine the Total Expected Value

Since the flips are independent, the expected value of the sum of heads in 10 flips, E(S), is given by \[ E(S) = 10 \times E(X) = 10 \times 0.5 = 5 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a measure of how likely an event is to happen. When flipping a fair coin, the probability of landing heads is 0.5, and the probability of landing tails is also 0.5.
We write this as: \( P(\text{Heads}) = 0.5 \) and \( P(\text{Tails}) = 0.5 \).
Probability values range between 0 and 1, where 0 means the event will not happen at all and 1 means the event will definitely happen.
To find the expected number of heads in 10 coin flips, we use this probability information. Each flip is independent, which means previous flips do not affect the outcome of the next flip.
Random Variable
A random variable is a variable that takes on different values based on the outcome of a random event. In our case, we let X be the random variable for one coin flip.
X can take on the value 1 if the coin lands heads, or 0 if it lands tails.
This simplifies our calculations.
For instance, in the solution, we use the fact that the expected value of X, \( E(X) \), is calculated as follows:
\[ E(X) = 0.5 \times 1 + 0.5 \times 0 = 0.5 \]
Here, the expected value represents the average outcome of X over a large number of trials. This value helps in determining the expected number of heads.
Independent Events
Two events are independent if the outcome of one event does not affect the outcome of the other.
If we flip a coin, the result of previous flips does not affect the result of the next flip. This is crucial when calculating expected values.
In the given problem, flipping the coin 10 times constitutes 10 independent events. Therefore, we can sum up the expected values for each flip to find the total expected number of heads.
Mathematically, this can be represented as:
\[ E(S) = \text{Number of flips} \times E(X) = 10 \times 0.5 = 5 \]
Thus, the expected number of heads when a fair coin is flipped 10 times is 5.

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Most popular questions from this chapter

What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Let \(X\) be the number appearing on the first die when two fair dice are rolled and let \(Y\) be the sum of the numbers appearing on the two dice. Show that \(E(X) E(Y) \neq E(X Y) .\)

In roulette, a wheel with 38 numbers is spun. Of these, 18 are red, and 18 are black. The other two numbers, which are neither black nor red, are 0 and \(00 .\) The probability that when the wheel is spun it lands on any particular number is 1\(/ 38\) . a) What is the probability that the wheel lands on a red number? b) What is the probability that the wheel lands on a black number twice in a row? c) What is the probability that the wheel lands on 0 or 00\(?\) do? d) What is the probability that in five spins the wheel never lands on either 0 or 00\(?\) e) What is the probability that the wheel lands on one of the first six integers on one spin, but does not land on any of them on the next spin?

What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads?

What is the probability that a player of a lottery wins the prize offered for correctly choosing five (but not six) numbers out of six integers chosen at random from the integers between 1 and \(40,\) inclusive?
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