The Tilted Board

Draw the full tree — n = 3 rows, p = 0.7.

At each peg the ball goes R (right) with probability 0.7, or L (left) with probability 0.3. Draw every possible path on your whiteboard. Label each branch with its probability.

Write the probability of every complete path as a product — e.g. the path RRL has probability \(0.7 \times 0.7 \times 0.3\). Do this for all 8 paths.

Look at the paths that land in the same bin.

Group your 8 paths by which bin they reach (bin 0 = all lefts, bin 3 = all rights). What do you notice about the probabilities of paths in the same group? Rewrite each path probability using exponents — e.g. \(p^2(1-p)^1\).

How many paths land in each bin? Write the four counts. Call the number of rights \(k\). Can you write the probability of one path to bin \(k\) using only \(k\), \(n\), \(p\)?

Write P(X = k) — then name the pieces.

The total probability of landing in bin \(k\) is: (number of paths to bin \(k\)) \(\times\) (probability of one such path). Write this for each \(k = 0, 1, 2, 3\). Verify the four values sum to 1.

What is the mathematical name for the number of paths to bin \(k\)? Write the general formula for \(P(X = k)\) in terms of \(n\), \(k\), and \(p\).

Extend to n = 4 — and look for the pattern in the counts.

Extend your tree by one row to n = 4. Count paths to each bin and write the five counts in a row beneath your n = 3 counts. Do the same for n = 2 and n = 1 above it.

What pattern connects each row to the one above? Can you predict the n = 5 row without drawing the tree?

The path count for bin \(k\) in an \(n\)-row board turns out to have a closed-form expression. You may have seen it before — how would you write it mathematically, using \(n\) and \(k\)?

Build the full probability table for \(X \sim B(4,\, 0.7)\).

For each value \(k = 0, 1, 2, 3, 4\), calculate \(P(X = k) = \binom{4}{k}(0.7)^k(0.3)^{4-k}\). Set up a table on your whiteboard with columns for \(k\), \(\binom{4}{k}\), \((0.7)^k\), \((0.3)^{4-k}\), and \(P(X=k)\). Verify the probabilities sum to 1.

Check against the board.

Go back to the simulator. Set n = 4, p = 0.7 and drop 500 balls. Does the histogram match your table? Which bin is tallest? Is that what you predicted?

When can you use this model?

Think about what had to be true about the Plinko board for your formula to work. List the four conditions that must hold for a situation to be modelled by a binomial distribution.

We write \(X \sim B(n, p)\) when those conditions are met. What does each symbol mean?

Investigate the mean — use the simulator.

Set n = 10. Drop 500 balls for each of the following values of p: 0.3, 0.5, and 0.7. Record the observed mean each time. What pattern do you notice? Write a conjecture for E(X) in terms of n and p.

Investigate the variance — then write the formula.

Keep n = 10. Vary p from 0.1 to 0.9 in steps of 0.2 and record the observed variance each time. At which value of p is the spread greatest? Why does that make sense — which tilt makes the outcome most unpredictable?

You should now have conjectures for both E(X) and Var(X). Write them as formulas in terms of n and p. Check your variance formula against each entry in your table.

Solve a problem — and use your GDC.

Dr. Nemo, a marine biologist, claims that 15% of clownfish in a reef are born with an unusual fin pattern. She observes 25 randomly selected clownfish, all from different families.

Let \(X\) be the number of clownfish with the unusual fin pattern. Write down the distribution of \(X\).

Find:   (i) \(P(X = 3)\)    (ii) \(P(X \leq 4)\)    (iii) \(P(X \geq 2)\)    (iv) \(\text{E}(X)\) and \(\text{Var}(X)\)

First attempt each part using your formula. Then verify using your GDC: