Math Problem Statement
An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of "heads" (h) and "tails" (t) which we write hth, ttt, etc. For each outcome, let N be the random variable counting the number of tails in each outcome. For example, if the outcome is tht, then =Ntht2. Suppose that the random variable X is defined in terms of N as follows: =X−N2−N4. The values of X are given in the table below. Outcome tth hth thh tht hht htt ttt hhh Value of X −2 −4 −4 −2 −4 −2 2 −4 Calculate the probabilities P=Xx of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value x of X P=Xx
Solution
To solve this problem, we need to determine the probabilities for each value of the random variable . The probability distribution depends on the frequency of occurrence of each value of .
Step-by-Step Solution:
1. List the possible outcomes of tossing the coin three times:
The outcomes of 3 coin tosses are combinations of heads (h) and tails (t). There are possible outcomes:
2. Identify the corresponding values of :
We are given the values of for each outcome:
\text{Outcome} & : \quad tth, hth, thh, tht, hht, htt, ttt, hhh \\ X & : \quad -2, -4, -4, -2, -4, -2, 2, -4 \end{aligned}$$ #### 3. **Determine the probabilities $$P(X = x)$$:** We now calculate the probability of each value of $$X$$ based on how often each value occurs in the list of outcomes. - The total number of outcomes is $$8$$. - The probability of each outcome is $$\frac{1}{8}$$ because the coin is fair and all outcomes are equally likely. Now, we can count the occurrences of each value of $$X$$ and calculate the probabilities. #### 4. **Count occurrences for each value of $$X$$:** - $$X = -2$$ appears for outcomes: $$tth, tht, htt$$. This happens 3 times, so $$P(X = -2) = \frac{3}{8}$$. - $$X = -4$$ appears for outcomes: $$hth, thh, hht, hhh$$. This happens 4 times, so $$P(X = -4) = \frac{4}{8} = \frac{1}{2}$$. - $$X = 2$$ appears for the outcome $$ttt$$. This happens 1 time, so $$P(X = 2) = \frac{1}{8}$$. #### 5. **Create the table for the probability distribution of $$X$$:** The table of values of $$X$$ and their corresponding probabilities $$P(X = x)$$ is: | Value $$x$$ of $$X$$ | $$-4$$ | $$-2$$ | $$2$$ | |--------------------|-------|-------|-----| | Probability $$P(X = x)$$ | $$\frac{1}{2}$$ | $$\frac{3}{8}$$ | $$\frac{1}{8}$$ | #### Final Answer: The probabilities of $$X = x$$ are: - $$P(X = -4) = \frac{1}{2}$$ - $$P(X = -2) = \frac{3}{8}$$ - $$P(X = 2) = \frac{1}{8}$$ Would you like any further details or clarifications on the calculation? Here are 5 related questions that may expand your understanding: 1. How would the probabilities change if the coin were biased? 2. What is the expected value of $$X$$ for this distribution? 3. How do we calculate the variance of $$X$$ in this case? 4. What if the coin is tossed 4 times instead? How would the random variable $$N$$ and $$X$$ be adjusted? 5. Can you explain how the law of large numbers would apply to repeated experiments of this coin toss? **Tip:** The sum of all probabilities in any probability distribution should always equal 1, which is a good way to check your work.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Random Variables
Discrete Probability Distribution
Formulas
Probability P(X = x) = Frequency of X / Total Outcomes
X = -N^2 - N^4
Theorems
Law of Total Probability
Concept of Expected Value
Suitable Grade Level
Grades 9-12