Materials & Prep
Gather two fair coins for the opening and display or copy the guided-practice and exit-ticket questions. No other special setup is needed.
Opening
Display two fair coins, labeled A and B. State that coin A has landed heads and ask students to predict whether coin B is more likely to land heads or tails. Students commit with a thumb vote and briefly explain. Establish that coin A does not affect coin B, so each outcome for coin B remains equally likely.
Then display the sequence H, H, H, H and ask students to predict the next result. Have them commit before discussion. Record the claim “tails is due” without correcting it yet. Connect: the four tosses are separate trials just as coins A and B are separate trials. Introduce the question for today: does what happened earlier physically change the next trial?
Direct instruction
DefineIndependent trials are trials in which the outcome of one trial does not change the possible outcomes or probabilities of another trial. For an ideal fair coin, every toss has P(heads) = 1/2 and P(tails) = 1/2.
Worked exampleFor the sequence H, H, H, H, calculate two different probabilities. From the beginning, the probability of getting five heads in a row is 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32. However, once the first four heads have already happened, the probability that the next toss is heads is still 1/2, and the probability that it is tails is still 1/2. The first probability describes an entire five-toss sequence. The second describes only the next toss.
Think aloudI must first identify the trial I am predicting. If it is only the next toss, I use the two equally likely outcomes of a fair coin. The common error is to say tails has probability 1 because four heads occurred. That treats the coin as if it remembers earlier tosses. A fair coin has no mechanism that makes tails more likely after heads.
ConfrontAsk, “If four heads make tails certain, what would make the fifth toss different from the first toss?” Students turn and talk. Emphasize that “tails is due” is a belief about balancing results, not a probability argument. A later outcome may balance the overall record, but it is never forced on an independent trial.
Guided practice
Students solve and justify each item with a partner. They must write the probability and complete the sentence frame, “The next trial is still because .”
- A fair die has shown 6, 6, and 6. What is P(6) on the next roll? Explain.
- A fair spinner has four equal sections: red, blue, green, and yellow. It has landed red four times. What is P(blue) on the next spin? Explain.
- A bag contains 3 red and 2 blue marbles. A marble is drawn, replaced, and the bag is mixed. After three red draws, what is P(red) on the next draw? Explain.
- A bag contains 3 red and 2 blue marbles. A red marble is drawn and not replaced. Is the next draw independent of the first? What is P(red) now?
ExpectedStudents should answer 1/6, 1/4, 3/5, and “not independent; P(red) = 2/4 = 1/2” respectively. Check for understanding: students show a fist for independent and an open hand for not independent as you read each situation. Ask one student to explain why replacement matters in item 3 but not item 4. If students say item 1 is 1 because six is due, return to the fair die’s six possible outcomes.
ScaffoldProvide a two-column organizer labeled “Did the earlier result change the materials?” and “Probability of the next outcome.” Let students use a picture of the coin, die, or bag and answer orally, by pointing, or in writing before completing the sentence frame.
Independent problem solving
Students complete these four problems independently, showing a probability and a one-sentence explanation.
- A fair coin shows tails seven times in a row. Find P(heads) on the next toss.
- A fair six-sided die shows 2, 2, 2, and 2. Find P(2) on the next roll.
- A spinner has 5 equal sections, 2 yellow and 3 purple. It has landed yellow three times. Find P(purple) on the next spin.
- Explain the difference between P(getting five heads in a row) and P(heads on the next toss after four heads). Use numbers.
SupportStudents who need an entry point may draw the possible outcomes for the next trial first and use the frame, “Earlier results do/do not change the next trial because the is/is not changed.”
ExtensionChallenge ready students to write a fair-coin sequence that is unlikely from the start but still has P(heads) = 1/2 on its next toss. They should explain why both statements can be true.
Closing
Exit ticket“A fair coin has landed heads four times. A student says, ‘The next toss must be tails.’ Explain why the student is incorrect. State P(tails) and P(heads) for the next toss, and state the probability of five heads in a row from the beginning.”
Look for responses that distinguish the next independent trial from the entire sequence: P(tails) = 1/2, P(heads) = 1/2, and P(five heads in a row) = 1/32. Use incorrect exit tickets to identify whether the student confuses ‘unlikely sequence’ with ‘changed next-trial probability.’