A coach claims that on the next league game the odds of her team winning are 3:1; the odds against losing are 5:1; and the odds against a tie are 7:1. Can these odds be right? Explain.
odds = probability yes / probability no
so
prob win/prob lose = 3/1
however we know that prob not win (lose or tie)= 1 - prob win
so
w /(1-w) = 3
w = 3 - 3w
w =3/4 = 0.75 = probabiity of win = P(W)
0.25 = prob lose or tie = P(L+T)
odds win or tie = 5/1
prob win or tie / prob lose = 5
P(L) = [P(W)+P(T)] / 5 = [.75 +P(T)] / 5
odds no tie = odds win or lose = 7/1
prob win or lose / prob tie = 7
P(T) = [P(W)+P(L)] / 7 = [ .75 + P(L)] /7
now P( W) + P(L) + P(T) must = 1
.75 + [.75 +P(T)] / 5 + [ .75 + P(L)] /7 = 1?
.75 ( 1 + .2 + .142) + .2 P(T) + .142 P(L) = 1 ?
1.0065 + .2 P(T) + .142 P(L) = 1 ?
Nope, not unless one of those probabilities is negative
To determine if the given odds are possible, we need to check if they are consistent with each other.
The odds of winning are stated as 3:1, which means that the team has a 3 out of 4 chance of winning.
The odds against losing are 5:1, indicating that the team has a 5 out of 6 chance of not losing or winning.
The odds against a tie are 7:1, suggesting that the team has a 7 out of 8 chance of not tying.
To check if these odds are consistent, we need to calculate the probability of each outcome (winning, not losing, not tying) and see if they add up to the total probability of all possible outcomes, which is 1.
The probability of winning is 3/4, the probability of not losing is 5/6, and the probability of not tying is 7/8.
To calculate the total probability, we multiply these probabilities: (3/4) * (5/6) * (7/8) = 105/192.
If the odds were correct, this total probability should equal 1, but 105/192 is not equal to 1.
Therefore, the given odds are not consistent or accurate.
To verify whether the given odds are consistent, we can perform a simple calculation. The odds for an event can be expressed as a ratio of the number of favorable outcomes to the number of unfavorable outcomes.
Let's start by calculating the odds of winning. According to the coach, the odds of winning are 3:1. This means that for every 3 favorable outcomes (winning), there is 1 unfavorable outcome (not winning). Therefore, the probability of winning can be calculated as:
Probability of winning = Favorable outcomes / (Favorable outcomes + Unfavorable outcomes)
= 3 / (3 + 1)
= 3/4
= 0.75 (or 75%)
Now let's calculate the odds against losing. The odds against losing are given as 5:1. This implies that for every 5 unfavorable outcomes (losing), there is 1 favorable outcome (not losing, which includes winning or tying). Therefore, the probability of not losing can be calculated as:
Probability of not losing = Favorable outcomes / (Favorable outcomes + Unfavorable outcomes)
= 1 / (5 + 1)
= 1/6
≈ 0.167 (or 16.7%)
Finally, let's calculate the odds against a tie. The odds against a tie are given as 7:1. This means that for every 7 unfavorable outcomes (a tie), there is 1 favorable outcome (not tying, which includes winning or losing). Therefore, the probability of not tying can be calculated as:
Probability of not tying = Favorable outcomes / (Favorable outcomes + Unfavorable outcomes)
= 1 / (7 + 1)
= 1/8
= 0.125 (or 12.5%)
Now, if we add up the probabilities of winning (0.75), not losing (0.167), and not tying (0.125), the total probability should equal to 1 (or 100%). Let's check if this holds true:
0.75 + 0.167 + 0.125 = 1.042
As we can see, the sum is greater than 1, which means the given odds cannot be accurate. In a valid scenario, the sum of probabilities should always equal 1.
Therefore, the coach's claim about the odds is not correct because the probabilities derived from the odds do not add up to 1.