In 1961, Richard Feynman began giving a two-year series of lectures to freshmen and sophomores at Caltech. His lectures covered nearly every facet of physics: from gravity to quantum mechanics and thermodynamics. One of his lectures was on probability and Pascal’s Triangle. Note that some of the ideas presented below were not stated by Feynman, but rather originate from Scott Carter.
Four combinations exist when a two-sided fair coin is flipped twice: TH, HT, HH, and TT. It can be seen, then, that the probability that the coin lands on heads once is 2/4, the probability that it lands on heads twice is 1/4, and the probability that it does not land on heads at all is 1/4. It is interesting to note that the series of numerators described by those probabilities is equal to the second row of Pascal’s Triangle:
1 2 1
1 3 3 1
1 4 6 4 1
It follows that when a coin is flipped three times, the probabilities reflect the third row of Pascal’s Triangle, namely 1,3,3,1 [the probabilities themselves are 1/8 for three heads, 3/8 for two heads and one tail, 3/8 for one head and two tails, and 1/8 for three tails]. The denominator is determined by the formula 2n where n is the number of flips. It turns out that these type of probabilities can be described by:
P = ( n K ) /2n
where n is the number of outcomes and k is the number of heads thrown.
This notation works well only when a two-sided object [like a coin] is used. However, what happens when one takes a vacation to Frogstar D and is expected to pay in three-sided units? It turns out that Pascal’s triangle can no longer be used to describe the nominators of the probabilities when one flips the three-sided coin multiple times. Instead, the Pascal’s 3-simplex must be used. Simply, this is Pascal’s triangle in in three dimensions. For instance, if I flipped a three-sided coin twice, there are nine possible outcomes [thus the denominator is 9]. The probabilities would look like this: 1/9 for both heads, 2/9 for one head and a tail, 2/9 for one head and an other, 1/9 for both tails, 2/9 for a tail and an other, and 1/9 for all others. The numerators, 1,2,1,2,1,2, describe the second level of Pascal’s 3-simplex. The equation for the probabilities when tossing a three sided coin is:
P = ( n K ) /3n
But what about higher dimensions? In higher dimensions one would substitute that dimension number in place of the 3 in the equation above [i.e. in four dimensions, substitute a 4 for the 3]. But one would also have to envision Pascal’s 4-simplex, which is, of course, in four dimensions. It gets a bit more difficult here!