Math Detours · Grade 5 · Around the corner · 14

Heads, heads, heads, heads

Flip a coin fifty times and write it down. Real luck has longer streaks than people expect, and that is how fakes get caught.

Name ____________________Date ____________

Tonight's idea

Flip a coin and it lands heads (H) or tails (T). Each is as likely as the other. A streak is the same side several times in a row, like H H H H.

One row below came from 50 flips of a real coin. A person made up the other, trying to look random. Which is which?

A B heads tails
Fifty flips each, in groups of ten.

Row B is the real one. It has a streak of six tails. Row A never has more than three the same in a row. That is the clue.

People think a long streak looks wrong, so they change sides too often. A real coin cannot remember its last flip. Flip 50 times, and about 8 lists in every 10 have a streak of five or more.

Real luck is lumpy. Long streaks are normal, and a list with none is suspicious.

Watch one: read a list of flips

Here are 20 flips: TTHHH THTTH HHHTT HHTHT

The longest streak is four heads. Now count the switches, the places where the side changes. There are 10.

Twenty flips have 19 gaps between them. A real coin switches at about half of the gaps, so 9 or 10 is what to expect. This list looks real.

Luck also evens out over many tries. Roll dice thousands of times at mathdetours.com/demos/dice.html.

Your turn

  1. Here are 20 flips: HTTTT THHTT HHHTT TTHHH (a) How many heads are there? (b) How long is the longest streak? (c) How many switches are there?
  2. Flip a coin 50 times. Write H or T after each flip. (a) How long is your longest streak? (b) How many heads did you get?
  3. Two flips can land in 4 ways: HH, HT, TH, TT. Only one way is two heads, so the chance of two heads is 1 in 4. (a) List all the ways three flips can land. What is the chance of three heads? (b) How many ways can four flips land? What is the chance of four heads?
  4. One of these lists of 30 flips is real and one is made up.
    X: HHTHT THTHH THHTH THTTH TTHTH HTHTT
    Y: TTHTT THHTH HTHHH TTHHH TTTTT HTHHH
    (a) Find the longest streak and the number of switches in each. (b) Which list is made up? Give two reasons.

Stretch: is tails due?

A coin has landed heads four times in a row. Your friend says tails is "due". What is the chance that the next flip is tails? Then explain why five heads in a row is still rarer than four.

Answers

  1. (a) 9 heads. (b) 5: the five tails near the start. (c) 6 switches.
  2. Your own flips. A longest streak of 4, 5, 6 or 7 is common. The number of heads is usually between 20 and 30.
  3. (a) HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. That is \(2 \times 2 \times 2 = 8\) ways, so three heads is 1 in 8. (b) \(2 \times 2 \times 2 \times 2 = 16\) ways, so four heads is 1 in 16.
  4. (a) X: longest streak 2, and 21 switches. Y: longest streak 5, and 13 switches. (b) X is made up. It has no streak longer than 2, which a real coin almost never manages in 30 flips. It also switches too often: 30 flips have 29 gaps, so about 14 or 15 switches is normal, not 21.

Stretch. 1 in 2, the same as always. The coin has no memory. Five heads is rarer because it needs four heads first and then one more, and that last flip comes up heads only half the time: 1 in 32 instead of 1 in 16.

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