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Solutions

Solutions#

Counting Principle#

TODO

Combinations#

TODO

Probability#

  1. TODO

  2. The word “atleast” is a red flag in problems involving probability. If you see the word “atleast”, it is a fair bet you will need to find the complement of a set at some point. To see why, note the way this problem is phrase can be interpretted with the Aristotle’s Square of Opposition. The Square of Opposition is pictured below for quick reference,

assignments/assets/imgs/sets/square_of_opposition.jpg

The Square tells us the complement of “some are” (“atleast one”) is “none are”. Therefore, we can express the probability this problem is seeking with the Law of Complements,

P(A) = 1 - P(A^c)

Where A^c corresponds to the event of getting no heads.

This is equivalent to getting all tails, which is a much simpler event to find, because it only has one outcome, namely ttt.

To calculate the probability of this event, we apply the The Fundamental Counting Principle to find the total number of ways the experiment can occur. Each flip has two outcomes, heads or tails. There are three flips in total. Thus,

n(S) = 2 \cdot 2 \cdot 2 = 8

The probability sought can then be calculated as,

P(A) = 1 - \frac{1}{8} = \frac{7}{8}

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