Suppose that a biased coin has a probability of 0.45 of coming up tails.If the coin is tossed three times, what is the probability of getting tails exactly twice?

To find the probability of getting tails exactly twice in three tosses of a biased coin, we can use the binomial probability formula.

The binomial probability formula is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

Where:

• P(X=k) is the probability of k successes in n trials
• C(n, k) is the number of combinations of n items taken k at a time
• p is the probability of success on any given trial
• n is the number of trials
• k is the number of successes

In this case, we want to find P(X=2), the probability of 2 tails in 3 tosses. We know that:

• n = 3 (the number of tosses)
• k = 2 (the number of tails we want)
• p = 0.45 (the probability of getting tails)

Step 1: Calculate C(n, k)

C(n, k) = C(3, 2) = 3! / [(3-2)! * 2!] = 3

Step 2: Calculate p^k

p^k = (0.45)^2 = 0.2025

Step 3: Calculate (1-p)^(n-k)

(1-p)^(n-k) = (1-0.45)^(3-2) = 0.55

Step 4: Substitute these values into the binomial probability formula

P(X=2) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 3 * 0.2025 * 0.55 = 0.3341 (rounded to four decimal places)

So, the probability of getting tails exactly twice in three tosses of the biased coin is approximately 0.3341 or 33.41%.