Discrete Probability – Flipping a coin 5 times in a row.

$ \begingroup $ There are two possibilities for each of the five tosses of the coin, so there are $ 2^5 = 32 $ possible outcomes in your sample outer space, as you found .

What is the probability that heads never occurs doubly in a rowing ?

Your proposed answer of $ 13/32 $ is decline.

If there are four or five heads in the succession of five coin tosses, at least two heads must be straight .
If there are three heads in the succession of five coin tosses, the lone possibility is that the sequence is HTHTH .
There are $ \binom { 5 } { 2 } = 10 $ sequences of five coin tosses with precisely two heads, of which four have straight heads ( since the first of these back-to-back heads must appear in one of the first four positions ). Hence, there are $ 10 – 4 = 6 $ sequences of five coin tosses with precisely two heads in which no two heads are consecutive .
In each of the five sequences of mint tosses in which precisely one pass appears, no two heads are consecutive .
In the entirely sequence of five coin tosses in which no heads appear, no two heads are straight.

Hence, the number of sequences of five coin tosses in which no two heads are consecutive is $ 0 + 0 + 1 + 6 + 5 + 1 = 13 $, as you found .

What is the probability that neither heads nor tail occurs doubly in a row ?

Your proposed answer of $ 1/16 $ is chastise since there are only two favorable cases : HTHTH and THTHT, which gives the probability $ \frac { 2 } { 32 } = \frac { 1 } { 16 } $ .

What is the probability that both heads and tails occur at least doubly in a row ?

Your proposed answer of $ 15/16 $ is faulty.

Since $ 1 – \frac { 1 } { 16 } = \frac { 15 } { 16 } $, your answer suggests you mistakenly believed that the negation of the affirmation that neither heads nor chase occurs twice in a row is that both heads and tails occur at least twice in a course. The negation of the statement that neither heads nor tail occurs doubly in a row is that at least two heads or at least two tails are back-to-back. For example, the sequences HHTHT and TTTTH both violate the restriction that neither heads nor tails occur doubly in a row without satisfying the stronger prerequisite that both heads and tails occur at least doubly in a row .
If both heads and tails occur at least twice in a course, then there are four possibilities :

  • there is a block of three consecutive heads and a block of two consecutive tails
  • there is a block of three consecutive tails and a block of two consecutive heads
  • there is a block of two consecutive heads and a single head that are separated by a block of two consecutive tails
  • there is a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads

A auction block of three back-to-back heads and a barricade of two consecutive tails can occur in two ways, HHHTT and TTHHH. By symmetry, a block of three straight tails and two straight heads can occur in two ways. A block of two consecutive heads and a individual head that are separated by a block of two consecutive tails can occur in two ways, HHTTH and HTTHH. By symmetry, a barricade of two consecutive tails and a unmarried fag end that are separated by a block of two straight heads can occur in two ways. Hence, there are $ 2 + 2 + 2 + 2 = 8 $ friendly cases, giving a probability of $ \frac { 8 } { 32 } = \frac { 1 } { 4 } $ .

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