Colour runs in a shuffled deck
A standard deck of 52 cards, 26 red and 26 black, is shuffled uniformly at random. A run is a maximal block of consecutive cards of the same colour: for example, red red black red has three runs.
Find the expected number of runs.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Every new run starts with a change of colour, except the first run.
Look at one pair of neighbouring cards. What is the probability that their colours differ?
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Count colour changes instead of runs: the number of runs is one more than the number of places where the colour changes. Then use one indicator per pair of neighbouring cards.
Step 1. Runs , where is the number of positions from 1 to 51 where card and card have different colours.
Step 2. Take positions and . The first card is red or black; given its colour, the second is one of the 51 remaining cards, 26 of which have the other colour. So the colours differ with probability .
Step 3. By linearity,
In general. With red and black cards, the expected number of colour changes is , so there are runs on average.