A random variable X is defined as the number of heads observed when a coin is tossed 4 times. Make a chart that shows the probability distribution for X. What is the expected value?
To create a probability distribution chart for the random variable X, which represents the number of heads observed when a coin is tossed 4 times, we need to consider all possible outcomes and their associated probabilities.
When a fair coin is tossed, there are two possible outcomes for each toss: heads (H) or tails (T). Since the coin is tossed four times, there are a total of 2^4 = 16 possible outcomes.
Let's calculate the probability for each value of X:
X = 0 (no heads): probability P(X = 0) = P(TTTT) = (1/2)^4 = 1/16
X = 1 (1 head): probability P(X = 1) = P(HTTT, THTT, TTHT, TTTH) = 4 * (1/2)^4 = 4/16 = 1/4
X = 2 (2 heads): probability P(X = 2) = P(HHTT, HTHT, HTTH, THHT, THTH, TTHH) = 6 * (1/2)^4 = 6/16 = 3/8
X = 3 (3 heads): probability P(X = 3) = P(HHHT, HTHH, THHH, HHTH, HTHH, THHH) = 4 * (1/2)^4 = 4/16 = 1/4
X = 4 (4 heads): probability P(X = 4) = P(HHHH) = (1/2)^4 = 1/16
Now, we can summarize the probability distribution for X:
X | 0 | 1 | 2 | 3 | 4
P(X) | 1/16 | 1/4 | 3/8 | 1/4 | 1/16
The expected value (mean) of a discrete random variable is calculated by multiplying each value by its corresponding probability and summing them up. Let's calculate the expected value of X:
Expected value (E[X]) = (0 * 1/16) + (1 * 1/4) + (2 * 3/8) + (3 * 1/4) + (4 * 1/16)
= 0 + 1/4 + 6/8 + 3/4 + 0
= 0 + 1/4 + 3/4 + 0
= 1/4 + 3/4
= 4/4
= 1
Therefore, the expected value of X is 1.
To create a chart showing the probability distribution for X, we need to calculate the probability of each outcome when a fair coin is tossed 4 times. Let's break down the problem step-by-step:
Step 1: Determine the sample space.
When a coin is tossed 4 times, there are 2 possible outcomes for each toss (heads or tails). Therefore, the sample space is 2^4 = 16.
Step 2: Calculate the probability for each outcome.
Since the coin is fair, the probability of getting heads on any given toss is 1/2, and the probability of getting tails is also 1/2. Since each toss is independent, we can multiply the probabilities of individual outcomes together.
To find the probability of each outcome, we'll consider the number of heads (X) in the 4 tosses:
- When X = 0 (no heads), there is only one possible outcome: TTTT. So, the probability is (1/2)^4 = 1/16.
- When X = 1 (one head), there are four possible outcomes: HTTT, THTT, TTHT, TTTH. So, the probability is 4 * (1/2)^4 = 4/16 = 1/4.
- When X = 2 (two heads), there are six possible outcomes: HHTT, HTHT, HTTH, THHT, THTH, TTHH. So, the probability is 6 * (1/2)^4 = 6/16 = 3/8.
- When X = 3 (three heads), there are four possible outcomes: HHHT, HHTH, HTHH, THHH. So, the probability is 4 * (1/2)^4 = 4/16 = 1/4.
- When X = 4 (four heads), there is only one possible outcome: HHHH. So, the probability is (1/2)^4 = 1/16.
Now, let's summarize the probability distribution for X:
X | 0 | 1 | 2 | 3 | 4
----------------------------------------------
P(X) | 1/16 | 1/4 | 3/8 | 1/4 | 1/16
Step 3: Calculate the expected value.
The expected value (also known as the mean or average) is calculated by taking the sum of each outcome multiplied by its corresponding probability.
Expected value (E(X)) = Σ(X * P(X))
Using the probability distribution we obtained earlier, the calculation is as follows:
E(X) = (0 * 1/16) + (1 * 1/4) + (2 * 3/8) + (3 * 1/4) + (4 * 1/16)
Simplifying the above equation yields:
E(X) = (0 + 1/4 + 6/8 + 3/4 + 0) = 5/2 = 2.5
Therefore, the expected value of X is 2.5.