# Deer Button Game Odds

Deer Button is a game played by people of Woodland Nations. Players use eight two-colour counters. Players take turns throwing all eight deer buttons at the same time. They win beans according to this table:

Number of Buttons of the same colour | Beans Awarded

8---------------------------------------------10
7---------------------------------------------4
6---------------------------------------------2
other----------------------------------------0

a) Determine the probability that a player will score 10 points on a given throw.

b) What is the probability of scoring at least 4 points in a throw?

Thanks!~

Last edited by a moderator:

#### DarnItJimImAnEngineer

I'll assume colours black (b) and red (r)
a) If we fictitiously number the buttons, the probability of #1 being red is 50 % (P(1,r) = 1/2). The probability of #2 being red is 50 %. â€¦
The probability of all eight being red is (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/256.
The probability of all eight being black is (1/2)^8 = 1/256.
So there are two possible throws out of 256 that score 10 points.
P = 2/256 = 1/128 â‰ˆ 0.78 %

I'll leave it to you to figure out b) from here.

^^^

That is what I came up with. The reason I posted is because this is from a Gr. 12 Statistics Textbook. The answer given at the back is:

a) 20% b) 40%

There are 5 possible outcomes to this game: 8 same, 7 same, 6 same, 5 same, or 4 same. If there are 3 the same, then there are 5 of the other colour, and so on. So, the probability of 8 buttons the same colour is 20%. Scoring at least 4 points means 7 or 8 buttons the same colour. So, the probability of 7 or 8 buttons the same colour is 40%.

:unsure::shock:

#### DarnItJimImAnEngineer

That's easy. The textbook author is an idiot who doesn't actually understand probability.

There are five possible outcomes, but they are not equally probable.

1 person

#### tahirimanov19

It is same with the coin toss. Let's assume colors are Aero Blue and Blizzard Blue (these are real color names). The assumption is buttons are fair. Therefore
$$Pr(X \; and \; Y) = Pr(X) \times Pr(Y)$$
$$Pr(All \; buttons \; are \; Aero \; Blue) = Pr(All \; buttons \; are \; Blizzard \; Blue) = 2^{-8}$$
$$Pr(All \; buttons \; are \; same \; color)=2 \times 2^{-8}=2^{-7}=0.0078125=0.78125 \text{%}$$
-------------------
$$Pr( \text{7 buttons are Aero Blue} \; AND \; \text{1 buttons is Blizzard Blue})=\frac{8}{256}= 2^{-5}$$
Same with reverse. So,
$$Pr( \text{Only seven buttons are same color} ) = 2^{-4}=0.0625=6.25 \text{%}$$
--------------------
$$Pr( \text{At least 4 points}) = Pr(All \; buttons \; are \; same \; color \; OR \; \text{Only seven buttons are same color}) = Pr(All \; buttons \; are \; same \; color) + Pr( \text{Only seven buttons are same color} ) = 0.0703125 = 7.03125 \text{%}$$

#### EvanJ

That's easy. The textbook author is an idiot who doesn't actually understand probability.

There are five possible outcomes, but they are not equally probable.
When books teach probability, they give examples with coins, dice, and cards that involve equally likely outcomes. I think it would be useful for books to say directly that outcomes do not have to be equally likely. I made up an exaggerated example of outcomes that are not equally likely. Whenever you get in your car, you can survive (probability of almost 100%) or die in a car accident (probability near 0%).

#### tahirimanov19

It is not hard to calculate. Think of it as a coin toss, where the outcome is either Heads (H) or Tails (T).
8 coins are tossed, which means there are $2^8=256$ outcomes.
And now calculate the number of versions of same outcomes:
All Heads have one version, and all Tails have one version. Therefore, the number of getting all heads or all tails = 2.
Now, getting 7 Heads and 1 Tails has 8 versions.
HHHHHHHT
HHHHHHTH
HHHHHTHH
HHHHTHHH
HHHTHHHH
HHTHHHHH
HTHHHHHH
THHHHHHH
Same with 7 Tails and one Heads.
Therefore, the number getting only 7coins with the same side=16

Now, getting at least 7 number of coins with same side = 16+2=18.

Divide the results by 256, and you get the probability.

!!Combinations!!

#### skipjack

Forum Staff
Where does the textbook state that each bean is worth one point?