Moderate -0.3 This is a straightforward expected value calculation requiring students to find probabilities from given constraints, calculate expected profit per roll, then multiply by 100. It involves only basic probability arithmetic and no conceptual challenges beyond understanding expected value, making it slightly easier than average for A-level.
6 A six-sided die is biased so that the probability of scoring 6 is 0.1 and the probabilities of scoring \(1,2,3,4\), and 5 are all equal. In a game at a fête, contestants pay \(\pounds 3\) to roll this die. If the score is 6 they receive \(\pounds 10\) back. If the score is 5 they receive \(\pounds 5\) back. Otherwise they receive no money back. Find the organiser's expected profit for 100 rolls of the die.
Can be implied, e.g. by 18; or using exp no. of 5's & 6's: \(18 \times 5\) or \(10 \times 10\)
\(3 \times 0.18\) or \(2 \times 0.18\) or \(7 \times 0.1\) (result of these)(poss \(\times 100\))
M1
\(5 \times 0.18\) or \(10 \times 0.1\)(result of these)(poss \(\times 100\)); (\(3 \times 0.18\) only scores if using £3, not score of 3. Similarly for \(2 \times 0.18\))
\(4 \times 3 \times 0.18\) AND \(2 \times 0.18 + 7 \times 0.1\) (poss \(\times 100\)); (or 2.16 AND 1.06 or 216 AND 106)
M1
3 AND \(5 \times 0.18 + 10 \times 0.1\) (poss \(\times 100\)); (or 3 AND 1.9 or 300 AND 190); NB \(300 + 100 \times 0.18 + 100 \times 0.1\) is insuff
\('2.16' - '1.06'\) or \('216' - '106'\); must be attempt gain on 1,2,3,4 – loss on 5,6
M1 dep any M1
\(3 - '1.9'\) or \(300 - '190'\); must be attempt receipt – payout on 5,6. Eg: \(300 - 100 \times (5 \times 0.18 + 6 \times 0.1) = 150\); M1M1M0M1A0
\(E(\text{profit for 100 rolls}) = (£)110\)
A1
Mark one method only; Must be matched pair; e.g. 300–106 or 216–190: M1M1M0M0A0
Total: [5]
# Question 6:
$(1 - 0.1) \div 5 = 0.18$ | M1 | Can be implied, e.g. by 18; or using exp no. of 5's & 6's: $18 \times 5$ or $10 \times 10$
$3 \times 0.18$ or $2 \times 0.18$ or $7 \times 0.1$ (result of these)(poss $\times 100$) | M1 | $5 \times 0.18$ or $10 \times 0.1$(result of these)(poss $\times 100$); ($3 \times 0.18$ only scores if using £3, not score of 3. Similarly for $2 \times 0.18$)
$4 \times 3 \times 0.18$ AND $2 \times 0.18 + 7 \times 0.1$ (poss $\times 100$); (or 2.16 AND 1.06 or 216 AND 106) | M1 | 3 AND $5 \times 0.18 + 10 \times 0.1$ (poss $\times 100$); (or 3 AND 1.9 or 300 AND 190); NB $300 + 100 \times 0.18 + 100 \times 0.1$ is insuff
$'2.16' - '1.06'$ or $'216' - '106'$; must be attempt gain on 1,2,3,4 – loss on 5,6 | M1 dep any M1 | $3 - '1.9'$ or $300 - '190'$; must be attempt receipt – payout on 5,6. Eg: $300 - 100 \times (5 \times 0.18 + 6 \times 0.1) = 150$; M1M1M0M1A0
$E(\text{profit for 100 rolls}) = (£)110$ | A1 | Mark one method only; Must be matched pair; e.g. 300–106 or 216–190: M1M1M0M0A0
**Total: [5]**
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6 A six-sided die is biased so that the probability of scoring 6 is 0.1 and the probabilities of scoring $1,2,3,4$, and 5 are all equal. In a game at a fête, contestants pay $\pounds 3$ to roll this die. If the score is 6 they receive $\pounds 10$ back. If the score is 5 they receive $\pounds 5$ back. Otherwise they receive no money back. Find the organiser's expected profit for 100 rolls of the die.
\hfill \mbox{\textit{OCR S1 2012 Q6 [5]}}