Problem:
Two biased coins C1 and C2 have probabilities of getting heads 2/3 and 3/4, respectively, when tossed. If both coins are tossed independently two times each, then what is the probability of getting exactly two heads out of these four tosses?

Solution:

To solve this problem, we can use the concept of probability and the rules of independent events.

Step 1: Determine the probabilities of getting two heads from each coin individually.

Let's calculate the probabilities of getting two heads from each coin individually.

For coin C1:
- The probability of getting a head on a single toss is 2/3.
- The probability of getting tails on a single toss is 1 - 2/3 = 1/3.

For coin C2:
- The probability of getting a head on a single toss is 3/4.
- The probability of getting tails on a single toss is 1 - 3/4 = 1/4.

Step 2: Determine the probabilities of getting two heads out of four tosses.

There are four tosses in total, and we want to find the probability of getting exactly two heads.

There are multiple ways to achieve this:
1. HHHT: Head, Head, Head, Tails
2. HHTH
3. HTHH
4. THHH

For each of these combinations, we need to calculate the probability.

For the combination HHHT:
- The probability of getting a head on the first toss (C1) is 2/3.
- The probability of getting a head on the second toss (C1) is 2/3.
- The probability of getting a head on the third toss (C1) is 2/3.
- The probability of getting tails on the fourth toss (C2) is 1/4.

Therefore, the probability of the combination HHHT is (2/3) * (2/3) * (2/3) * (1/4).

Similarly, we can calculate the probabilities for the other combinations: HHTH, HTHH, and THHH.

Step 3: Calculate the total probability.

To find the total probability of getting exactly two heads out of four tosses, we need to sum up the probabilities of all the possible combinations.

Total probability = Probability of HHHT + Probability of HHTH + Probability of HTHH + Probability of THHH.

After calculating the probabilities for each combination, we can add them up to find the total probability.

Step 4: Calculate the final probability.

After adding up the probabilities, we get the final probability of getting exactly two heads out of four tosses using biased coins C1 and C2.

This final probability can be expressed as a fraction or a decimal, depending on the given values.

Conclusion:
Using the concept of probability and the rules of independent events, we can find the probability of getting exactly two heads out of four tosses using biased coins C1 and C2. The final probability can be calculated by summing up the probabilities of all the possible combinations.