"A players shuffles 7 red cards and 3 green cards into a random order then places them in a line. If all green cards are net to each other, he wins £10. If all are apart, he wins £2. Otherwise he wins nothing. Find the price up to which the player would be willing to expect a profit."
Please help, I've been on this question for a long time and cannot find a breakthrough.
To find the price up to which the player would be willing to expect a profit, we need to calculate the expected value for each potential outcome.
Let's start by analyzing the different scenarios:
1. All green cards are together:
There are 8 possible positions for the block of green cards within the line of 10 cards (7+1, since the green cards can be adjacent to any of the red cards). Given that the cards are shuffled randomly, each position has an equal probability of occurring.
The probability of all green cards being together is 8/10 = 4/5.
In this case, the player wins £10.
2. All green cards are apart:
There are 8 possible positions for the green cards to be placed independently within the line of 10 cards (7+2, since the green cards need to be separated by at least one red card). Given that the cards are shuffled randomly, each position has an equal probability of occurring.
The probability of all green cards being apart is 8/10 = 4/5.
In this case, the player wins £2.
3. Green cards are neither all together nor all apart:
There are 10! possible orders in which the 10 cards can be arranged (10 factorial). Out of these, the 8 positions where the green cards are together and the 8 positions where the green cards are apart are already counted in the previous scenarios.
Therefore, the remaining possible positions where the green cards are neither all together nor all apart are (10! - 8 - 8).
The probability of the green cards being neither all together nor all apart is ((10! - 8 - 8) / 10!) = ((10! - 16) / 10!).
Now, let's calculate the expected value:
Expected Value = (Probability of Green Cards Together * Winnings for Green Cards Together) + (Probability of Green Cards Apart * Winnings for Green Cards Apart) + (Probability of Neither Together nor Apart * Winnings for Neither Together nor Apart)
Expected Value = ((4/5) * £10) + ((4/5) * £2) + (((10! - 16) / 10!) * £0)
Simplifying the equation, we get:
Expected Value = (£8) + (£1.6) + (£0)
Expected Value = £9.60
Thus, the price up to which the player would be willing to expect a profit is £9.60.