While staying in a 15-story hotel, Polya plays the following game. She enters an elevator on the 6th floor. She flips a fair coin five times to determine her next five stops. Each time she flips "heads, " she goes up one floor. Each time she flips "tails, " she goes down one floor. What is the probability that each of her next five stops is on the 7th floor or higher? Express your answer as a common fraction.

Answer:The probability is [tex]\frac{1}{2}[/tex]Step-by-step explanation:We are going to modeled this problem from the number of heads.Let be H : ''The number of heads after five flips''H can assume the following values : h = 0h = 1 h = 2 h = 3h = 4h = 5 H ~ Bi (n,p)H ~ Bi (5,0.5)H can be modeled as a Binomial random variable.In which p = 0.5 is the success probability (the probability of head in one flip of a fair coin)n = 5 is the number of times that we flip the coin.We are also assuming independence in each flip.The probability function for H is [tex]P(H=h)=(nCh)p^{h}(1-p)^{n-h}[/tex][tex]P(H=h)=(5Ch)0.5^{h}0.5^{5-h}[/tex]For all the possible values of H H ≥ 3 is the event that puts Polya on the 7th floor or higher.Now we calculate the probability of H ≥ 3[tex]P(H\geq 3)=P(H=3)+P(H=4)+P(H=5)[/tex][tex]P(H\geq 3)=(5C3)0.5^{3}0.5^{2}+(5C4)0.5^{4}0.5^{1}+(5C5)0.5^{5}0.5^{0}[/tex][tex]P(H\geq 3)=0.3125+0.15625+0.03125=0.5[/tex][tex]P(H\geq 3)=\frac{1}{2}[/tex]Therefore, the probability of each of her next five stops is on the 7th floor or higher is [tex]\frac{1}{2}[/tex]