15 Barney has a biased coin. When the coin is thrown once, the probability that tho coin will land heads is 0.3 Bamey throws the coin 4 times. (a) Work out the probability that the coin will land heads exactly 3 times. (b) Work out the probability that the coin will lard heads at least once. (b) Coblity

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To solve the problem, we will use the binomial probability formula, which is given by:

$$
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
$$

where:
- $$ n $$ is the number of trials (coin tosses),
- $$ k $$ is the number of successful outcomes (heads),
- $$ p $$ is the probability of success on a single trial,
- $$ \binom{n}{k} $$ is the binomial coefficient, calculated as $$ \frac{n!}{k!(n-k)!} $$.

### Given:
- Probability of heads, $$ p = 0.3 $$
- Probability of tails, $$ 1 - p = 0.7 $$
- Number of trials, $$ n = 4 $$

### (a) Probability of getting exactly 3 heads

We need to find $$ P(X = 3) $$:

$$
P(X = 3) = \binom{4}{3} (0.3)^3 (0.7)^{4-3}
$$

Calculating the binomial coefficient:

$$
\binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4 \times 3!}{3! \times 1!} = 4
$$

Now substituting into the formula:

$$
P(X = 3) = 4 \times (0.3)^3 \times (0.7)^1
$$

Calculating $$ (0.3)^3 $$ and $$ (0.7)^1 $$:

$$
(0.3)^3 = 0.027
$$
$$
(0.7)^1 = 0.7
$$

Now substituting these values back:

$$
P(X = 3) = 4 \times 0.027 \times 0.7
$$

Calculating this:

$$
P(X = 3) = 4 \times 0.0189 = 0.0756
$$

### (b) Probability of getting at least 1 head

To find the probability of getting at least one head, we can use the complement rule:

$$
P(\text{at least 1 head}) = 1 - P(\text{no heads})
$$

First, we calculate $$ P(X = 0) $$:

$$
P(X = 0) = \binom{4}{0} (0.3)^0 (0.7)^{4}
$$

Calculating the binomial coefficient:

$$
\binom{4}{0} = 1
$$

Now substituting into the formula:

$$
P(X = 0) = 1 \times (0.3)^0 \times (0.7)^4
$$

Calculating $$ (0.7)^4 $$:

$$
(0.7)^4 = 0.2401
$$

Thus,

$$
P(X = 0) = 0.2401
$$

Now, using the complement rule:

$$
P(\text{at least 1 head}) = 1 - P(X = 0) = 1 - 0.2401 = 0.7599
$$

### Final Answers:
(a) The probability that the coin will land heads exactly 3 times is $$ 0.0756 $$.

(b) The probability that the coin will land heads at least once is $$ 0.7599 $$.
(a) The probability of getting exactly 3 heads is 0.0756. (b) The probability of getting at least one head is 0.7599.

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