Tuesday

"The start-table game," Comet says. "Flip two coins. The runner with the most heads leads the warm-up lap."

"Four equally likely outcomes," Wren says, writing HH, HT, TH, TT. "X is the number of heads."

"Zero heads once, one head twice, two heads once," Comet counts. "1/4, 1/2, 1/4. Expected value 1."

"That one came from the sample space," Wren says. "Now the practice laps. No sample space tells us how many laps a runner does."

Nova projects Tuesday's tally. "Would you like a hint? Use the counts as the probabilities."

"6 ran one lap, 10 ran two, 4 ran three, out of 20," Comet reads. "3/10, 1/2, 1/5."

"Expected value 1.9 laps," Wren says. "What do you notice? Same method, different source."

"Theoretical from counting outcomes, empirical from a tally," Comet says. "Both make a distribution."

Two coin flips have 4 equally likely outcomes: HH, HT, TH, TT. X counts the heads.

Group the outcomes by value. X = 0: TT. X = 1: HT and TH. X = 2: HH.

Each probability is a group count over 4: 1/4, 1/2, 1/4. These are theoretical probabilities.

Expected value: 0 × 1/4 + 1 × 1/2 + 2 × 1/4 = 1 head.

Laps per runner at Tuesday practice have no sample space to count. The crew uses Nova's tally of 20 runners instead.

P(1 lap) = 6/20 = 3/10, P(2 laps) = 1/2, P(3 laps) = 1/5. These are empirical probabilities.

Expected value: 1 × 3/10 + 2 × 1/2 + 3 × 1/5 = 1.9 laps per runner.

For 30 runners on Race Day, the crew expects about 1.9 × 30 = 57 laps, so about 57 water cups.

Same method both ways: list the values, attach probabilities, check they add to 1, then multiply and add. Only the source of the probabilities differs.

Two sources, one method. Tomorrow is Data Lab: twenty draws from the pennant bag against the bag's own distribution.