Hey guys, so ns think the very first thing I"m walk to execute is write out all of the possibilities (Sample Space). Yet basically since we"re flipping the coin 5 times, and also there are only 2 possibilities, there should be 2^5 = 32 outcomes (size the Sample Space)?

Also, how deserve to I create out all the possibilities come ensure that i am not missing any? due to the fact that they said "By counting" for this reason I"m assuming they want us to write every little thing out in the Sample Space, and also then count it out.

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There space two possibilities because that each of the five tosses that the coin, for this reason there room $2^5 = 32$ feasible outcomes in your sample space, together you found.

What is the probability the heads never ever occurs double in a row?

Your proposed prize of $13/32$ is correct.

If over there are four or 5 heads in the sequence of five coin tosses, at least two heads should be consecutive.

If there are three heads in the succession of 5 coin tosses, the only possibility is that the succession is HTHTH.

There room $inom52 = 10$ assignment of 5 coin tosses with precisely two heads, the which 4 have consecutive top (since the very first of these consecutive top must appear in among the first four positions). Hence, there space $10 - 4 = 6$ sequences of 5 coin tosses with exactly two heads in which no two heads are consecutive.

In every of the five sequences the coin tosses in which exactly one head appears, no 2 heads are consecutive.

In the only sequence of 5 coin tosses in i beg your pardon no top appear, no 2 heads space consecutive.

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

What is the probability the neither heads nor tails occurs twice in a row?

Your proposed answer of $1/16$ is correct because there are just two favorable cases: HTHTH and also THTHT, which offers the probability $frac232 = frac116$.

What is the probability the both heads and tails take place at the very least twice in a row?

Your proposed price of $15/16$ is incorrect.

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Since $1 - frac116 = frac1516$, her answer says you mistakenly thought that the negative of the statement that neither heads no one tails occurs double in a row is the both heads and also tails happen at least twice in a row. The negative of the statement that neither heads nor tails occurs twice in a heat is the at least two top or at least two tails room consecutive. For instance, the sequences HHTHT and TTTTH both violate the restriction that neither heads no one tails occur twice in a row without to solve the stronger necessity that both heads and also tails take place at least twice in a row.

If both heads and tails take place at the very least twice in a row, then there are 4 possibilities:

there is a block of 3 consecutive heads and also a block of two consecutive tailsthere is a block of three consecutive tails and a block of two consecutive headsthere is a block of two consecutive heads and a solitary head that are separated by a block of 2 consecutive tailsthere is a block of two consecutive tails and a single tail that space separated by a block of two consecutive top

A block of 3 consecutive heads and also a block of 2 consecutive tails can happen in 2 ways, HHHTT and also TTHHH. Through symmetry, a block of three consecutive tails and also two consecutive heads can occur in two ways. A block of 2 consecutive heads and a single head that room separated through a block of 2 consecutive tails can happen in 2 ways, HHTTH and also HTTHH. By symmetry, a block of 2 consecutive tails and also a single tail that room separated by a block of 2 consecutive top can occur in two ways. Hence, there are $2 + 2 + 2 + 2 = 8$ favorable cases, offering a probability that $frac832 = frac14$.