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Statistics 9-12 / Week 12 / Monday
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Week 12 · Random Variables, Expected Value and Race Day

Monday

A random variable and its distribution
// The awards table, built from expected value
⏱ about 20 min

Monday: A Random Variable and Its Distribution

"Every finisher draws one token from the pennant bag," Comet says. "5 marked 1, 3 marked 2, 2 marked 3."

"So the number of pennants is a random variable," Wren says. "A number for each token. Call it X."

"P(X = 1) is 1/2, P(X = 2) is 3/10, P(X = 3) is 1/5," Comet reads. "They add to 1."

Nova projects three bars. "Would you like a hint? Graph it like a data set, values across and probabilities up."

"What do you notice?" Wren asks. "The bars lean left. Most finishers take one pennant."

"Then how many pennants should we make for 40 finishers?" Comet asks. "We cannot make exactly the right number."

"We can find the mean," Wren says. "Each value times its probability, added. 1.7 pennants per finisher, in the long run."

"Race Day is Saturday," Comet says. "Let us build the whole awards table on that."

Race Day: pennants and runners on the loop; Comet holds a medal tray beside an expected-value chart as Wren draws names.

A random variable

A random variable gives each outcome a number. Here X is the pennants on the drawn token: 1, 2 or 3.

Its probability distribution lists each value with its probability. 5 of the 10 tokens are marked 1, so P(X = 1) = 1/2.

Every number this week is the crew's own reading from Nova's Run Log, not a fact about any real race.

Pennants, XTokensProbability
151/2
233/10
321/5
A bar chart of the pennant distribution: values 1, 2 and 3 with probabilities 1/2, 3/10, 1/5.

The distribution is graphed like a data set: values across, probabilities up. The bars add to 1 instead of to a count.

A solved problem to study

Here is how Wren finds the expected value, the mean of the distribution.

  1. Multiply each value by its probability: 1 × 1/2, 2 × 3/10, 3 × 1/5.
  2. That gives 1/2, 3/5 and 3/5.
  3. Add them: 17/10 = 1.7.
  4. Read it back: over many draws, the mean is 1.7 pennants per finisher. No single token shows 1.7.
  5. For 40 finishers, expect about 1.7 × 40 = 68 pennants in all.

Expected value is a weighted mean. Common values weigh more because their probabilities are larger.

THE PENNANT DISTRIBUTION
  • Read the question.
  • Tap your answer.
A bar chart of a probability distribution: values 1, 2, 3 with probabilities 1/2, 3/10, 1/5A token shows 1 pennant with probability 1/2, 2 with probability 3/10, 3 with probability 1/5. What is the expected value, in pennants?
What is P(X is 2 or more), the probability a finisher takes home at least 2 pennants?
The expected value is 1.7 pennants per finisher. For 40 finishers, about how many pennants should the crew make?
The expected value is 1.7 pennants. What does that number mean?
How many tokens in the pennant bag are marked 2? Type the number.
WHY THIS EXERCISEThat count over the total is P(X = 2). Every probability in the distribution starts from a count.
StatementTrue or false?
1/2 + 3/10 + 1/5 = 1?
A random variable gives each outcome a number.?
An expected value must be one of the values the variable can take.?
1 × 1/2 + 2 × 3/10 + 3 × 1/5 = 1.7?
WHY THIS EXERCISEThe distribution and its mean are the two things every random variable has.
Try it
Put 10 scraps in a cup: 5 marked 1, 3 marked 2, 2 marked 3.
Draw one, write it down, put it back. Do it ten times and find the mean of your ten numbers.
Draw the pennant distribution as three bars. Mark the expected value on the value axis.

A strong start. Tomorrow two kinds of probability feed a distribution: counted from a sample space, or taken from a tally.