Basic Probability

See Probability for a recap of probability.

Intuition

1. An unbiased coin is flipped twice. Given that at least one of the tosses is a head, what is the chance that the other is a head?

Analysing this, we have two events:

• $A=$ at least one head
• $B=$ both are heads

It should be clear that these events affect each other and therefore are not independent.
So it should make sense that the answer might not be $\frac{1}{2}$.

We can calculate the answer ($P(B|A)$) like so:

$A=\{HH,TH,HT\}.$
$B=\{HH\}.$

$P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{P(\{HH\})}{P(\{HH,TH,HT\})}=\frac{1}{3}.$

2. Similarly, consider the following: An unbiased coin is flipped twice. Given that at least one of the tosses is a head, what is the chance that the other is a tail?

$A=$ at least one head
$B=$ the other is a tail

Evidently, these events also affect each other and therefore are not independent.
Again, it should then make sense if the answer is not $\frac{1}{2}$.
It may very well be, but let’s see what the answer is.

$A=\{HH,TH,HT\}.$
$B=\{TH,HT\}.$

$P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{P(\{TH,HT\})}{P(\{HH,TH,HT\})}=\frac{2}{3}.$