tmua.me Extreme Set A Paper 2 · Question 3

Each of $a_0,a_1,\ldots,a_6$ is chosen independently, with $1$ and $-1$ equally likely. Define

P(x)=x^7+a_6x^6+a_5x^5+\cdots+a_1x+a_0.

A student tries to find the probability that $P$ has at least one integer root.

If an integer $m$ is a root, then $m$ divides $a_0$ . Thus the only possible integer roots are $1$ and $-1$ .
$P(1)=0$ precisely when four of the seven chosen coefficients are $-1$ . Its probability is $\binom74/2^7=35/128$ .
The probability that $P(-1)=0$ is also $35/128$ .
Since the coefficients were chosen independently, the events $P(1)=0$ and $P(-1)=0$ are independent.
The required probability is therefore $2(35/128)-(35/128)^2$ .

Which of the following correctly identifies the first error and gives the required probability?

AThe first error is in I; the probability is

\dfrac{13}{32}

BThe first error is in IV; the probability is

\dfrac{13}{32}

CThe first error is in II; the probability is

\dfrac{35}{64}

DThe first error is in IV; the probability is

\dfrac{35}{64}

EThe first error is in V; the probability is

\dfrac{13}{32}

FThere is no error in the argument.

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