Probability formulation question!!?

Question

Using coin-flips only, design a game between two people in which the probability of winning for one of them is

(a) 1/7

(b) 3/7

(c) 5/7

Answer 1

Toss the coin 3 times - this results in the following possible results:
0 heads, p=1/8
1 head, p=3/8
2 heads, p=3/8
3 heads, p=1/8.

If you state as a rule that if 3 heads show up you discard and toss again, then your probabilities become
0 heads, p=1/7
1 head, p=3/7
2 heads, p=3/7
(These are the conditional probabilities given that you don't toss 3 heads.)

So your first two answers are:
(a) Toss 3 coins. If all 3 tosses are heads, discard and toss again until you get less than 3 heads. You win if the result is 3 tails.

(b) Toss 3 coins. If all 3 tosses are heads, discard and toss again until you get less than 3 heads. You win if the result is exactly one head (out of 3).

For (c), you can further refine the conditional probabilities for the ordered results HHH, HTT, THT, TTH, HHT, HTH, and THH. Just pick 5 out of these 7 equally probable events to define your winning conditions.

Another, completely different, way of going about it is to translate the toss sequence into a binary decimal. This effectively generates a uniform variable between zero and one. I win if the result is less than (a) 1/7, (b) 3/7 or (c) 5/7. For example, the sequence HHTHTT... becomes .110100... (binary) which is equal to .8125. You don't need to toss coins forever because you'll reach a point where you have enough precision to conclude that the result is below or above the threshold (for example HH = 0.11 = 3/4 = defitinely above 5/7 so I lose in all 3 cases after just two tosses). This will work for any probability that you want to achieve so it works better as a general solution.

Answer 2

Take three coins.

Paint one RED (R), one Blue (B), and one Green (G). -- this is for illustrative purposes only -- it could be any three colors.

Don't make the coat of paint so thick that it throws off probability or makes heads indistinguishable from tails.

For each game, there are two players - You and the person flipping the coins.

To remove bias, the chance of the "flipper" winning will be determined by your choice in the game.

You choose an outcome based on the rules in each of the three games, and the "flipper" flips the coins. If your result comes up, you win. If it does not, the "flipper" wins.
***FOR EACH GAME, WHEN ALL THREE COINS COME UP TAILS, DISREGARD THE RESULT AND RE-FLIP THE THREE COINS.***

The above rule will diminish the probability of any unique result to 1/7.

Now, Game #1:

Probability of winning:
YOU - 1/7
FLIPPER - 6/7

Flip all three coins. If ALL THREE come up HEADS, you win.

there are 8 combinations:

RBG
HHH -- 1/7
HHT -- 1/7
HTH -- 1/7
HTT -- 1/7
THH -- 1/7
THT -- 1/7
TTH -- 1/7
TTT -- REFLIP

Game #2:

Probability of winning:
YOU - 3/7
FLIPPER - 4/7

Choose TWO colors. If both of those colored coins come up the same (i.e.- BOTH heads or BOTH tails) You win.

[[SEE ABOVE FOR COMBINATIONS]]

There are 4 results out of eight - two where both coins will come up heads, and two where both coins will come up tails. Since one of the combinations for tails requires that ALL THREE coins come up tails, there are really only three combinations, since that specific combination results in a re-flip.

Game #3:

Probability of winning:
YOU - 2/7
FLIPPER - 5/7

Choose TWO colors. If both of those colored coins come up HEADS, you win.

This is the same as above, but with the single "double tails" outcome removed.

--These are just some of many of the outcomes, and arguably three of the simplest. You can adjust it many ways. Just look at the list of outcomes above, and you will be able to find some of your own patterns.

Answer 3

These types of question are all the same:
The game is that you have 3 coin flips, but if all the coins are tails, the outcome is undecided and the game starts anew.

Here are three possible ways (there are many):
(a) A wins if all the coins come up heads
(b) A wins if there is exactly one head
(c) A wins if either of the second two flips come up tails