The probability of obtaining “tails” when a biased coin is tossed is \(0.57\). The coin is tossed ten times. Find the probability of obtaining at least four tails.

The probability of obtaining “tails” when a biased coin is tossed is 0.57. The coin is tossed ten times. Find the probability of obtaining the fourth tail on the tenth toss.

evidence of recognizing binomial distribution     (M1)

e.g. \(X \sim {\rm{B}}(10,0.57)\) , \(p = 0.57\) , \(q = 0.43\)

EITHER

\({\rm{P}}(X \le 3) = 2.16 \times {10^{ - 4}} + 0.00286 + 0.01709 + 0.06041\) \(( = 0.08057)\)     (A1)

evidence of using complement     (M1)

e.g. \(1 - \) any probability,  \({\rm{P}}(X \ge 4) = 1 - {\rm{P}}(X \le 3)\)

\(0.919423 \ldots \)

\({\rm{P}}(X \ge 4) = 0.919\)     A1     N3

OR

summing the probabilities from \(X = 4\) to \(X = 10\)     (M1)

correct expression or values     (A1)

e.g. \(\sum\limits_{r = 4}^{10} {\left( {\begin{array}{*{20}{c}}
{10}\\
r
\end{array}} \right)} {(0.57)^r}{(0.43)^{10 - r}}\) , \(0.14013 + 0.2229 +  \ldots  + 0.02731 + 0.00362\)

0.919424

\({\rm{P}}(X \ge 4) = 0.919\)     A1     N3

[4 marks]

evidence of valid approach     (M1)

 e.g. three tails in nine tosses, \(\left( {\begin{array}{*{20}{c}}
9\\
3
\end{array}} \right){(0.57)^3}{(0.43)^6}\)

 correct calculation

 e.g. \(\left( {\begin{array}{*{20}{c}}
9\\
3
\end{array}} \right){(0.57)^3}{(0.43)^6} \times 0.57\) , \(0.09834 \times 0.57\)     (A1)

\(0.05605178 \ldots \)

 \({\text{P(4th tail on 10th toss)}} = 0.0561\)     A1     N2

[3 marks]

This was an accessible problem that created some difficulties for candidates. Most were able to recognize the binomial nature of the problem but were confused by the phrase "at least four tails" which was often interpreted as the complement of four or less. Poor algebraic manipulation also led to unnecessary errors that the calculator approach would have avoided.

This was an accessible problem that created some difficulties for candidates. Most were able to recognize the binomial nature of the problem but were confused by the phrase "at least four tails" which was often interpreted as the complement of four or less. Poor algebraic manipulation also led to unnecessary errors that the calculator approach would have avoided.