Intro to Probability - YAQ Week 3

Question

Let $n\leq n\in\mathbb{N}$ and show that $$ {m+1\choose n}={m\choose n}+{m\choose n-1}. $$

This identity can be verified by simply writing out the definition of each term and carrying out the computations. There is, however, a more conceptual way to understand it.
Take the $m+1$ objects we are choosing from and place $m$ of them together and separate the remaining one. Now we can choose $n$ objects from the pile of $m$ to get all ${m\choose n}$ possibilities that do not include the one object that we separated. The possibilities that are left over at this point will all contain the separated object and, thus, to get to $n$, we only need to choose $n-1$ objects from the "pile" of $m$, which gives ${m\choose n-1}$ possibilities and, eventually the sum on the right-hand-side of the identity. Of course this gives all possible ${m+1\choose n}$ ways to choose $n$ objects from $m+1$.

Question

What is the probability to have a straght in a five card hand? A straight occurs when the 5 cards are of consecutive values but not all of the same suit (in that case the hand would be called a straight flush).

Notice that if a hand consists of 5 consecutive cards, it can be uniquely described by the card of lowest value (we could equally well take the highest value). As we need 4 cards to follow the lowest, the latter can be at most a 10. We therefore have 10 possible choices for the value of the lowest card. Once that is determined we still can choose the suit of the five cards. This would give a total of $4^5$ possibilities but we need to exclude the 4 possible straight flushes, one of each suit, and thus the count is actually $4^5-4$. This yields a total of $10*(4^5-4)$ and a probability of $$ P(``\text{a randomly chosen hand is a straight"})=\frac{10(4^5-4)}{{52\choose 5}}=0.0039 $$

Question

Show the validity of the identity $$ n\binom{n}{i}=i\binom{n-1}{i-1} $$ for $1\leq i\leq n$ and $n\in\mathbb{N}$