A toy train can produce one of two musical sounds with equal probability on each trial. What is the probability that the same sound would be produced five times in a row?

A toy train can produce one of two musical sounds with equal probability on each trial. What is the probability that the same sound would be produced five times in a row?

• A. 1/32
• B. 11/32
• C. 1/16
• D. 11/16

Answer: (A)

This question tests understanding of independent trials with two equally likely outcomes and counting all favorable patterns. Each trial has two sounds, each with probability 1/2, and we want the same sound five times in a row. There are two favorable patterns: all five sounds are the first type, or all five are the second type. Each specific five-in-a-row pattern has probability (1/2)^5 = 1/32, so together they total 2 × 1/32 = 1/16. A quick alternative view: fix the first sound, then the next four must match it, giving (1/2)^4 = 1/16. The given option 1/32 would be the chance of five repeats of a particular sound, not of either sound.

This question tests understanding of independent trials with two equally likely outcomes and counting all favorable patterns. Each trial has two sounds, each with probability 1/2, and we want the same sound five times in a row. There are two favorable patterns: all five sounds are the first type, or all five are the second type. Each specific five-in-a-row pattern has probability (1/2)^5 = 1/32, so together they total 2 × 1/32 = 1/16. A quick alternative view: fix the first sound, then the next four must match it, giving (1/2)^4 = 1/16. The given option 1/32 would be the chance of five repeats of a particular sound, not of either sound.