This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Alright wajahat — let's do this. Step 1: Identify the sample space. When a fair die is rolled once, the possible outcomes for the up-turned face are the integers from 1 to 6. The sample space S = \1, 2, 3, 4, 5, 6\. The total number of possible outcomes is 6. Step 2: Define the random variable. Let X be the random variable representing the number on the up-turned face. The possible values for X are x \1, 2, 3, 4, 5, 6\. Step 3: Calculate the probability for each outcome. Since the die is fair, each outcome has an equal probability of occurring. The probability of any specific outcome x is given by: P(X=x) = Number of favorable outcomesTotal number of outcomes For each value of x from 1 to 6, there is only one favorable outcome (e.g., rolling a 1, rolling a 2, etc.). Therefore, for each x \1, 2, 3, 4, 5, 6\: P(X=x) = (1)/(6) Step 4: Present the probability distribution. The probability distribution can be shown in a table: |c|c| x & P(X=x) \\ 1 & (1)/(6) \\ 2 & (1)/(6) \\ 3 & (1)/(6) \\ 4 & (1)/(6) \\ 5 & (1)/(6) \\ 6 & (1)/(6) \\ The sum of all probabilities is (1)/(6) × 6 = 1, which confirms it is a valid probability distribution. The probability distribution for the up-turned faces is: P(X=x) = (1)/(6) for x \1, 2, 3, 4, 5, 6\ Send me the next one 📸