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A man flips a fair coin with sides heads and tails five times. Given that the man receives heads on at least two of the coin flips, what is the probability that he receives tails exactly twice after the five flips of the coin?

A.$\frac{1}{4}$ B.$\frac{5}{16}$ C.$\frac{5}{13}$ D.$\frac{3}{8}$ E.$\frac{8}{13}$

This process is memoryless and we have a $p=\dfrac{1}{2}$

What happened on two flips after five flips is the same as what happens on any two flips.

I think the answer is A but answer key says C.

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    (Not answering the question) I wonder, if you believe that memoryless is applicable, why wouldn't you consider the required probability the same as receiving tails exactly twice after three flips of the coin, i.e. D. $3/8$? – peterwhy Aug 12 '22 at 01:58
  • I think it's poor phrasing, and the question means to ask about the probability exactly two of the same five flips are tails. Possibly using "after" in the sense of "After the five flips are performed, the total he received" – aschepler Aug 12 '22 at 01:58
  • This may be similar to the Boy or Girl paradox, but in this case "Mr. Smith has five children. At least two of them are boys. What is the probability that exactly two children are girls?" – peterwhy Aug 12 '22 at 02:21
  • Previous coins problem and children problem. – peterwhy Aug 12 '22 at 02:32
  • As I read it, this is a trick question and the answer is 1/4. The trick is to force you to notice that when you have independent events, what has been seen so far does not matter for the future. – William M. Aug 12 '22 at 03:24
  • Another interpretation is let $X$ the number of heads. You want $P(X = 2 \mid X \geq 2).$ In this case you get $P(X=2)/(P(X=2)+P(X=3)+P(X=4)+P(X=5))=\ldots=\dfrac{5}{13}.$ – William M. Aug 12 '22 at 03:27
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    @WilliamM. He said that the answer key gives 5/13 in the question, so the answer is not 1/4. – Suzu Hirose Aug 12 '22 at 03:57
  • @SuzuHirose many times probability and or statistic questions are paradoxes soleley because of the wording rather than the mathematics (the mathematics does not reaches contradictions). The wording can be interpreted in several different ways. REGARDLESS of what the answer was, my first reading was what I said, and I've even written books in the subject. (I wouldn't know where this material came from but a lot of basic material is written by non-mathematicians in a more "interpret me" fashion than in a "here is a correctly unambiguous statement" one.) – William M. Aug 12 '22 at 14:48
  • @SuzuHirose In fact, statisticians do no write mathematics (and statistics is not mathematics!) Rather, they write statements of real problems and they want to somehow translate them into mathematics, this obviously creates many different, plausible and sensible, approaches. This is easily reflected in the fact that there are many dozen of overlapping algorithms for the same task or that other problems (e.g. Behrens-Fisher probem) have largely been unanswered and the existing solutions are not widely accepted (because the different interpretations require different approaches). – William M. Aug 12 '22 at 14:53

2 Answers2

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Given that here should be interpreted as In the condition that, which is a case of Conditional Probability, quoting Wikipedia:

In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred.

We define

  • $A$ to be the event that the man receives exactly two tails.
  • $B$ to be the event that he gets more than one head.

So the probability of getting at least two heads on 5 flips is

$$ P(B)=1 - \left(\frac 12 \right)^5-\binom{5}{1}\left(\frac 12 \right)^5 = \frac{13}{16} $$

While getting exactly two tails as well as getting at least two heads is

$$ P(A\cap B) = \binom{5}{2}\left(\frac 12 \right)^5 = \frac{5}{16} $$

Note: This value is the same as $P(A)$ because having exactly two tails implies that he's got at least two heads (three, actually).

Using the formula for conditional probability, that is,

$$ P(A\mid B) = \frac{P(A\cap B)}{P(B)} $$

We get,

$$ P(A\mid B) = \frac{\frac{5}{16}}{\frac{13}{16}} = \frac 5{13} $$

Which is what we wanted.

If $P(A|B) = P(A)$, then $A$ and $B$ must be independent. That is, knowledge about either event does not alter the likelihood of each other. Explained in English: As you already know that there are at least $2$ tails, the probability of it still having $2$ heads is changed relevant to not knowing anything.

Lily White
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  • I think this is the correct answer as to what the ambiguous question means. – Suzu Hirose Aug 12 '22 at 03:57
  • Why $1 - \left(\frac 12 \right)^5$? @LilyWhite –  Aug 12 '22 at 11:44
  • @BoredStar We are calculating the complementary because there are only two scenarios: no heads and 1 head, $\left(\frac 12\right)^5$ means heads on none of the tosses, which (fairly) should be written as $\left(1 - \frac 12\right)^5$ to be clearer. – Lily White Aug 12 '22 at 12:06
  • What is $P(A \cap B)$?@ LilyWhite –  Aug 12 '22 at 13:04
  • @BoredStar Wikipedia is your friend: https://en.wikipedia.org/wiki/Probability#Theory – Lily White Aug 12 '22 at 13:26
  • Where is $P(A \cap B)$ in your answer? $P(A|B) \neq \dfrac{5}{16}/\dfrac{13}{16}$ –  Aug 12 '22 at 13:30
  • @BoredStar My fault, it should be $P(A\cap B)$ there. The trick is that having exactly two tails implies that he's got three ($\ge 2$!) heads. – Lily White Aug 12 '22 at 14:23
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A man flips a fair coin with sides heads and tails five times. Given that the man receives heads on at least two of the coin flips, what is the probability that he receives tails exactly twice after the five flips of the coin?

This is poorly phrased.   My reading is that there are no further flips; the question is in regard to the result of these five flips.

To rephrase: "After the five flips: what is the probability that exactly two tails we obtained given that at least two heads were obtained."

Why is this coin-flip not memoryless and independent of past coin-flips for a fair coin, so we have to use bayes theorem and conditional probability?

They are the same five flips, not independent trials.

So we seek: "The probability for obtaining exactly two tails, given that at most three tails were among the five."

And thus Bayes' Rule is the appropriate to find the answer of $\underline{\phantom{5/13}}$.

Graham Kemp
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