Chances of getting exactly one heads in two coin flips?

Question

I know the chances of getting exactly one "heads" in two coin flips is 50%, as there are only four possible outcomes: HH, HT, TH, TT; and only two of those four meet the restraint I've placed. To figure this out, I think you can use a binomial distribution: 2C1 times 1/2 times 1/2 gives the probability of 1 in 2 that I'm looking for.

But why doesn't the formula $$P(A \text{ or }B) = P(A) + P(B) - P(A \text{ and } B)$$ (where A and B are heads on the first and second attempts, respectively) work? I get 75% (1/2 + 1/2 -1/4) which would work if I wanted the restraint "at least one 'heads,'" or simply what are the chances of getting one "heads."

Moreover, if you were to find the chances of getting "heads," wouldn't you take 1 minus the probability of getting two "tails" (25%) to get the 75% you wanted? So when would you use P(A or B)?

Answer 1

The symmetric difference, of exclusive or, is the probability of getting exactly one head. What you were calculating was the union, which is the probability of at least one head.

$\newcommand{\P}{\mathbb{P}}$In mathematics, "or" means "either $A$ is true, $B$ is true, or both $A$ and $B$ are true. See this Wikipedia page which has nice diagrams: https://en.wikipedia.org/wiki/Logical_disjunction.

In other words $\mathbb{P}(A\text{ or }B)= \mathbb{P}(A\text{ only})+ \P(B\text{ only})+ \P(\text{both }A\text{ and }B)$.

If I am interpreting your question correctly, $A = \{HH, HT \}$ and $B = \{ HH, TH\}$.

This means that $$A\text{ or }B = (A \text{ is true}) \lor (B \text{ is true}) = A \cup B = \{HH, HT, TH \}$$

Therefore $\P(A \text{ or }B) = \frac{3}{4}$ and the formula which you claim does not work does work. $$\P(A)=\frac{1}{2}, \quad \P(B)=\frac{1}{2}, \quad P(A \cap B)=\frac{1}{4}, \quad \frac{1}{2}+\frac{1}{2} - \frac{1}{4} = \frac{3}{4}$$

The word "or" is often used in English to mean "exclusive or" or "XOR", which means "either $A$ only, or $B$ only, but not both". This corresponds to the set operation of symmetric difference, rather than union. However, in probability, it is always union, and not symmetric difference, that is meant when the term "or" is used. This makes most mathematical expressions and statements much easier to state and formulate.

Here is a picture below of symmetric difference; contrast it with that of union you see above:

This means that $$ A \text{ xor }B = [(A\text{ is true})\lor(B\text{ is true})]\land ((A\text{ and }B)\text{ is not true}) \\= A \bigtriangleup B = \{HT, TH \} = A \cup B \setminus A \cap B = \{HH, HT, TH \} \setminus \{HH \}$$

Answer 2

No. See what. You look seperatly. If you define A as "the probability that only the first toss is head" its outcome probability will be 1/4, rather than 1/2. If you want to look the way you mentioned, than we will have the following $P(A)=P(the first is head)=\frac{1}{2}$ ; $P(B)=P(the second is head)=\frac{1}{2}$ , and $P(A&B) = P(both the first and th second tosses give head) =\frac{1}{4}$ (as it is one case of four). And we will have $P(A or B) = \frac{1}{2}+\frac{1}{2} - \frac{1}{4}$. But $P(A or B) = P( either first or second (or both) toss is head)$. See however that this does not exclude the situation when both are head. More precisely "(A or B) = (A and not B) + (B and not A)".

Answer 3

None of these "solutions" is correct because when a coin is flipped 2 times, there ARE NOT 4 possible outcomes. There are only 3 possible outcomes: 2 heads, 2 tails, or 1 head and 1 tail. The answer is 5 out of 12, with the probability of getting exactly 2 heads (or exactly 2 tails) 7 out of 24. A statistician (long since deceased) calculated this for me, but unfortunately did not share his calculations with me. The answer can be verified by flipping a coin a large number of times (100 times is sufficient, but the more flips the more precise) and recording the results. Next record the results of consecutive "pairs" of results i.e. HH, HT, TH,TT which will confirm that 1 head and 1 tail occurs 41.6% of the time.