13. james has organised a game to raise money...
13. james has organised a game to raise money for charity at a local fair. he rolls a fair six sided dice and flips a fair coin. if the coin lands on heads, the number on the dice is squared. if the coin lands on tails, the number on the dice is cubed. each person pays 50p to play. if they score above 30, they win £1. the game is played 450 times. how much money should james raise for charity?
Answer
# Explanation:
## Step1: Calculate the probability - coin and dice outcomes
The probability of getting heads on the coin flip is $P(H)=\frac{1}{2}$, and the probability of getting tails is $P(T)=\frac{1}{2}$. For the six - sided die, the possible outcomes are $1,2,3,4,5,6$.
## Step2: Calculate the scores for heads case
When the coin lands on heads, the score is the square of the die value.
For $n = 1$, $1^2=1$; for $n = 2$, $2^2 = 4$; for $n=3$, $3^2=9$; for $n = 4$, $4^2=16$; for $n=5$, $5^2 = 25$; for $n=6$, $6^2=36$.
The number of cases where the score is above 30 when the coin is heads is 1 (when the die shows 6). So the probability of getting a score above 30 when the coin is heads is $\frac{1}{6}$.
The combined probability of getting heads and a score above 30 is $P(H)\times\frac{1}{6}=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}$.
## Step3: Calculate the scores for tails case
When the coin lands on tails, the score is the cube of the die value.
For $n = 1$, $1^3=1$; for $n = 2$, $2^3 = 8$; for $n=3$, $3^3=27$; for $n = 4$, $4^3=64$; for $n=5$, $5^3 = 125$; for $n=6$, $6^3=216$.
The number of cases where the score is above 30 when the coin is tails is 3 (when the die shows 4, 5, 6). So the probability of getting a score above 30 when the coin is tails is $\frac{3}{6}=\frac{1}{2}$.
The combined probability of getting tails and a score above 30 is $P(T)\times\frac{1}{2}=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.
## Step4: Calculate the total probability of winning
The total probability of winning $P(W)=\frac{1}{12}+\frac{1}{4}=\frac{1 + 3}{12}=\frac{4}{12}=\frac{1}{3}$.
## Step5: Calculate the number of winners
The game is played 450 times. The number of winners is $n_w=450\times\frac{1}{3}=150$.
## Step6: Calculate the money received from players
Each person pays 50p or £0.5 to play. The total money received from 450 players is $450\times0.5=\text{£}225$.
## Step7: Calculate the money paid out to winners
Each winner gets £1. The total money paid out to 150 winners is $150\times1=\text{£}150$.
## Step8: Calculate the money raised for charity
The money raised for charity is the money received minus the money paid out, which is $225 - 150=\text{£}75$.
# Answer:
£75