SPS SPS SM 2021 February — Question 5 10 marks

Exam BoardSPS
ModuleSPS SM (SPS SM)
Year2021
SessionFebruary
Marks10
TopicBinomial Distribution
TypeE(X) and Var(X) with probability calculations
DifficultyEasy -1.3 This is a straightforward binomial distribution question requiring only direct application of standard formulas (mean = np, variance = np(1-p)) and calculator/table use for probabilities. All parts are routine textbook exercises with no problem-solving or conceptual insight required beyond recognizing the binomial model.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities
Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day. [1 mark]
    2. Find the variance of the number of times he falls off in a day. [1 mark]
    1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
    2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
  3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
    2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]
Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick.

Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard.

Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

\begin{enumerate}[label=(\alph*)]
\item 
\begin{enumerate}[label=(\roman*)]
\item Find the mean number of times he falls off in a day.
[1 mark]

\item Find the variance of the number of times he falls off in a day.
[1 mark]
\end{enumerate}

\item 
\begin{enumerate}[label=(\roman*)]
\item Find the probability that, on a particular day, he falls off exactly 10 times.
[2 marks]

\item Find the probability that, on a particular day, he falls off 5 or more times.
[3 marks]
\end{enumerate}

\item Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
\begin{enumerate}[label=(\roman*)]
\item Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
[2 marks]

\item Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.
[1 mark]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM 2021 Q5 [10]}}
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