In this question, I used a different method so I want to understand where am I going wrong:
Step 1: HT = 1/4 or TH = 1/4 Step 2: H + T = 1 Step 3: Substituting the value of T from step 1 which is T = 1/4H in step 2 Step 4: Solving for H with an equation that looks like:
H + 1/4H = 1
(4H^2 + 1)/4H = 1
4H^2 -4H + 1 = 0
H = 1/2
After solving for H, I am getting both the values of H as 1/2 so I chose option A which is incorrect but I want to understand what is incorrect in this solution? I understand that the question mentions that this is a biased coin but mathematically where am I going wrong?
Why have you multiplied ( P(H) x P(T)) with 2 ? To reflect the two situations of HT or TH?Additionally, could you break it down for me in easier terms why there is no direct proportionality? I am unable to understand. Sorry
You’re correct that the multiplication by 2 accounts for the two arrangements: HT and TH.
Regarding the proportionality: the key issue is that going the way you have and saying -
P(T) = (1/4) * P(H)
assumes a fixed ratio between the probabilities of heads and tails, which isn’t justified by the problem.
The problem doesn’t provide direct proportionality; it only states that the combined probability of HT or TH is 1/4.
In addition, a way to fact check your work - the question says the coin is biased, yet your math brought you to a probability of 1/2 for H (unbiased)
Those two things contradict each other. It should show you you’re going in the wrong direction.
Thank you. One reason also could be that my logic assumes that probability of T depends entirely on H because of the relationship but T must also individually be a valid probability but your explanation makes more sense to me. Thank you!
