Analyzing Random Walk Binomial Probabilities

00:00 8 marks

Consider a one-dimensional random walk where at each step a particle moves either left or right with equal probability. Let \(n\) be an integer representing the number of steps.

(a) Express the probability of the particle being at a position corresponding to exactly \(k\) right steps using a binomial coefficient.

(b) Expand the binomial expression and interpret the significance of each term in relation to the random walk outcomes.

Part a

5 minutes 4 marks

Question

Express the probability for being at a position with exactly \(k\) right steps after \(n\) steps in a random walk using the formula \(P(k) = \binom{n}{k}(\frac{1}{2})^n\). Explain your reasoning.

Hint Hint

Hint

Consider that each step has two equally likely outcomes and use the combination formula to select the rightward steps.

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Complete Answer

The probability is given by \(P(k) = \binom{n}{k}(\frac{1}{2})^n\), since there are \(\binom{n}{k}\) ways to choose \(k\) right steps out of \(n\) and each sequence occurs with probability \((\frac{1}{2})^n\).

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Part b

5 minutes 4 marks

Question

Expand the binomial expression \((p+q)^n\) and interpret the significance of each term, especially when \(p = q = \frac{1}{2}\).

Hint Hint

Hint

Recall that the expanded form is a sum of terms \(\binom{n}{k}p^kq^{n-k}\).

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Complete Answer

Expanding \((p+q)^n\) yields \(\sum_{k=0}^{n} \binom{n}{k}p^kq^{n-k}\). When \(p = q = \frac{1}{2}\), each term \(\binom{n}{k}(\frac{1}{2})^n\) represents the probability of obtaining exactly \(k\) right steps.

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