The bias of a coin (i.e., the probability of Heads) can take three possible values, 1/4 , 1/2 , or 3/4 , and is modeled as a discrete random variable Q with PMF
pQ(q)=⎧⎩⎨⎪⎪⎪⎪⎪⎪1/6,2/6,3/6,0,if q=1/4,if q=2/4,if q=3/4,otherwise.
Let K be the total number of Heads in two independent tosses of the coin. Find pQ|K(3/4|2) .
3/4
Luli's answer is wrong
Well, well, well, let's see here! We are trying to find the probability of having a bias of 3/4 given that we got 2 heads. Sounds like a mathematical mystery, doesn't it?
Now, to solve this riddle, we need to use Bayes' theorem. Remember, Bayes' theorem is like a secret code that helps us unravel the probability mysteries!
According to Bayes' theorem, pQ|K(3/4|2) is equal to pQ(3/4) times pK|Q(2|3/4), divided by pK(2).
Now, let's break it down. We know that pQ(3/4) is 3 out of 6, because our coin can take three possible values and pQ(3/4) is 3/6.
Next, we need to find pK|Q(2|3/4), which is the probability of getting 2 heads, given that the bias is 3/4. Since the coin tosses are independent, the probability of getting 2 heads in two tosses is (3/4)^2 = 9/16.
Lastly, we need to find pK(2), which is the probability of getting 2 heads regardless of the bias. This can happen in three different scenarios: bias of 1/4 and 3/4, or two tosses of 1/2. So pK(2) = (1/6 + 3/6 + 1/6) = 5/6.
Now, we put it all together:
pQ|K(3/4|2) = pQ(3/4) * pK|Q(2|3/4) / pK(2)
= (3/6) * (9/16) / (5/6)
= (3/6) * (9/16) * (6/5)
= 9/80
Therefore, my mathematical magician friend, the probability of having a bias of 3/4 given that we got 2 heads is 9/80. Voila!
To find pQ|K(3/4|2), we need to calculate the conditional probability of the bias being 3/4 (Q=3/4) given that there are 2 heads in two independent tosses of the coin (K=2).
Conditional probability can be calculated using the formula:
pQ|K(q|k) = pQ,K(q,k) / pK(k),
where pQ,K(q,k) is the joint probability of Q=q and K=k, and pK(k) is the probability of K=k.
Let's calculate each of these probabilities step by step:
1. Calculate pK(k):
We know that there are two independent tosses of the coin, and we want to find the probability of getting 2 heads. The possible outcomes for 2 tosses are {HH, HT, TH, TT}. To get 2 heads, we have only one outcome, HH. So pK(2) = 1/4.
2. Calculate pQ,K(q,k):
We need to find the joint probability of Q=q and K=k, which means the probability of having bias q and obtaining k heads in two tosses. We can calculate it by multiplying the probability of bias q with the conditional probability of getting k heads given bias q.
For pQ,K(3/4,2), we multiply pQ(3/4) with the probability of getting 2 heads given the bias of 3/4. From the given PMF, we can see that pQ(3/4) = 3/6 = 1/2.
Since the bias 3/4 means the probability of getting heads is 3/4, the probability of getting 2 heads in two tosses is (3/4)^2 = 9/16.
Therefore, pQ,K(3/4,2) = (1/2) * (9/16) = 9/32.
3. Calculate pQ|K(3/4|2):
Finally, we can calculate the conditional probability using the formula mentioned above:
pQ|K(3/4|2) = pQ,K(3/4,2) / pK(2) = (9/32) / (1/4) = (9/32) * (4/1) = 9/8.
So, pQ|K(3/4|2) = 9/8.