Probability distributions Revision Notes

In any simple probability scenario, the first step is to list every possible outcome that can occur. This is essential because probabilities are assigned to these outcomes, and none should be omitted. For example, when rolling a fair six-sided die, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. Each outcome is mutually exclusive, meaning only one can occur at a time, and collectively exhaustive, meaning one of them must occur.

• List outcomes carefully to avoid missing any.
• Ensure outcomes are mutually exclusive and collectively exhaustive.

Once all outcomes are identified, calculate the probability of each. The probability of an outcome is the chance that it will occur, expressed as a number between 0 and 1. For equally likely outcomes, the probability is 1 divided by the number of outcomes. For example, with a fair six-sided die, each outcome has probability 1/6 ≈ 0.1667. If outcomes are not equally likely, probabilities must be assigned based on the context or given data, ensuring they are valid (between 0 and 1).

• Probability of an outcome = (Number of favourable outcomes) / (Total number of possible outcomes) for equally likely cases.
• Probabilities must be between 0 and 1 inclusive.

A fundamental property of probability distributions is that the sum of the probabilities of all possible outcomes must be exactly 1. This reflects the certainty that one of the possible outcomes will occur. After calculating individual probabilities, add them together to check this condition. If the sum is not 1, re-examine the calculations or the list of outcomes for errors.

• Sum all probabilities: P(outcome 1) + P(outcome 2) +... + P(outcome n) = 1.
• If sum ≠ 1, check for missing outcomes or calculation mistakes.

To communicate the probabilities clearly, organise the outcomes and their probabilities into a probability distribution table. This table has two columns: one for the outcomes and one for their corresponding probabilities. This format makes it easier to interpret and use the probabilities for further calculations or analysis. For example, the distribution for a fair six-sided die is:

| Outcome | Probability |\n|---------|-------------|\n| 1 | 1/6 |\n| 2 | 1/6 |\n| 3 | 1/6 |\n| 4 | 1/6 |\n| 5 | 1/6 |\n| 6 | 1/6 |

• Use a two-column table: outcomes and probabilities.
• Ensure probabilities are expressed as fractions, decimals, or percentages consistently.
• Check that the table covers all outcomes and sums to 1.

Consider tossing a fair coin twice. The possible outcomes are: HH (both heads), HT (head then tail), TH (tail then head), and TT (both tails). Each outcome is equally likely with probability 1/4.

Step 1: List outcomes: HH, HT, TH, TT.\nStep 2: Calculate probabilities: each is 1/4 = 0.25.\nStep 3: Sum probabilities: 0.25 + 0.25 + 0.25 + 0.25 = 1.\nStep 4: Present in a table:

| Outcome | Probability |\n|---------|-------------|\n| HH | 0.25 |\n| HT | 0.25 |\n| TH | 0.25 |\n| TT | 0.25 |

• List all outcomes explicitly.
• Calculate and verify probabilities.
• Display in a clear table format.

Expected value is a fundamental concept in probability that represents the average or mean outcome of a random variable if an experiment is repeated many times. It is not necessarily a value the variable can take but rather a theoretical long-run average weighted by the probabilities of each outcome.

For example, if you roll a fair six-sided die many times, the expected value of the number shown is the average you would expect over a large number of rolls. This helps in predicting outcomes and making decisions under uncertainty.

• Expected value is a weighted average of all possible outcomes.
• Weights are the probabilities associated with each outcome.
• It provides a measure of central tendency for a probability distribution.

To calculate the expected value (denoted as E or μ) of a discrete random variable, follow these steps:

1. List all possible values the variable can take.

2. Identify the probability associated with each value; probabilities must sum to 1 and be between 0 and 1 inclusive. 3. Multiply each value by its corresponding probability to find the weighted value. 4. Sum all these weighted values to find the expected value.

• Formula: E = Σ (x × P(x)) where x is a value and P(x) its probability.
• Ensure probabilities sum to 1 before calculating.
• Expected value may not be one of the possible outcomes.

Consider a fair six-sided die with outcomes 1, 2, 3, 4, 5, and 6, each with probability 1/6.

Calculate the expected value: E = (1 × 1/6) + (2 × 1/6) + (3 × 1/6) + (4 × 1/6) + (5 × 1/6) + (6 × 1/6)

= (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 21 × 1/6 = 3.5. This means that over many rolls, the average outcome tends towards 3.5, even though 3.5 is not a possible roll value.

• Multiply each outcome by its probability.
• Sum all products to get expected value.
• Interpret result as long-run average outcome.

Expected value helps in decision-making and predictions where outcomes have different probabilities and values. For example, in games involving money or points, expected value can indicate the average gain or loss per play.

Example: A game costs £2 to play. You roll a fair die; if you get a 6, you win £10; otherwise, you win nothing. Calculate the expected net gain.

Step 1: List outcomes and probabilities: Win £10 with probability 1/6; win £0 with probability 5/6. Step 2: Calculate expected winnings: E = (10 × 1/6) + (0 × 5/6) = 10/6 ≈ £1.67. Step 3: Calculate expected net gain: Expected winnings - cost = £1.67 - £2 = -£0.33. This means on average, you lose 33p per game in the long run.

• Use expected value to assess fairness or profitability.
• Subtract costs to find expected net gain or loss.
• Interpret results in context to guide decisions.

Expected value is a theoretical average, not a guaranteed outcome.

Probabilities must be valid (between 0 and 1, summing to 1).

Expected value can be used to compare different scenarios or strategies in uncertain situations.

• Check probabilities before calculating.
• Interpret expected value carefully in context.
• Use expected value to inform decisions involving risk.