# Lego blocks and probability

You are given a Lego block of length 10 (length is measured discretely as number of “knobs”; the width of all blocks is assumed to be one) and three more Lego blocks of length 3, 4, 5, respectively. The Lego blocks are randomly stacked upon the block of length 10 such that their boundaries are contained within the boundaries of the block of length 10. The position of a short block is given as the position of the left-most knob of the 10-block covered by it. Hence, the block of length 3 could be with equal probability in position 1,2,3,4,5,6,7, or 8, the block of length 4 with equal probability in positions 1,2,3,4,5,6, or 7 and the block of length 5 with equal probabilty in positions 1,2,3,4,5 or 6. The short blocks are placed independently at random.

a. Describe and count the outcomes leading to position 2 respectively position 5 covered by all three blocks.

b. Let X be the number of knobs out of the ten of the large block, which are covered by all three shorter blocks; X ∈ {0, 1, 2, 3}. What is E[X]? Hint: Use indicator random variables.

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