Question 1209944
Let's break down each case using Bayesian inference:

**Likelihood:**
The likelihood of observing 8 heads in 10 flips is given by the binomial distribution:
$$L(p|X=8) = \binom{10}{8} p^8 (1-p)^2 = 45 p^8 (1-p)^2$$

**(a) No prior knowledge about 𝑝.**
* **Prior:** Uniform distribution over [0, 1], i.e., $P(p) \propto 1$.
* **Posterior:** $P(p|X=8) \propto L(p|X=8) \times P(p) \propto p^8 (1-p)^2$.
* **Estimated 𝑝:** To find the maximum, we differentiate and set to zero:
    * $\frac{d}{dp} [p^8 (1-p)^2] = 0$
    * This leads to $p = 8/10 = 0.8$.
* **(1) Estimated 𝑝 = 0.8**
* **(2) Yes, 𝑝 > 0.6**

**(b) The coin is likely fair (𝑝 around 0.5).**
* **Prior:** Beta distribution centered around 0.5, e.g., Beta(5, 5), so $P(p) \propto p^4(1-p)^4$.
    * For the sake of simplicity I will use Beta(6,6), which is also centered at 0.5.
* **Posterior:** $P(p|X=8) \propto p^8 (1-p)^2 \times p^5 (1-p)^5 = p^{13} (1-p)^7$.
* **Estimated 𝑝:** Differentiate and set to zero:
    * $\frac{d}{dp} [p^{13} (1-p)^7] = 0$
    * This leads to $p = 13/20 = 0.65$.
* **(1) Estimated 𝑝 = 0.65**
* **(2) Yes, 𝑝 > 0.6**

**(c) The coin is likely biased (𝑝 around 0 or 1).**
* **Prior:** Beta distribution biased towards 0 or 1, e.g., Beta(9, 1). So $P(p) \propto p^8(1-p)^0$.
* **Posterior** $P(p|X=8) \propto p^8(1-p)^2 * p^8 = p^{16}(1-p)^2$.
* **Estimated p**
    * $\frac{d}{dp}(p^{16}(1-p)^2) = 16p^{15}(1-p)^2-2p^{16}(1-p) = 0$
    * $16(1-p)-2p = 0$
    * $16-18p = 0$
    * $p= 16/18 = 8/9 \approx 0.888$.
* **(1) Estimated 𝑝 ≈ 0.888**
* **(2) Yes, 𝑝 > 0.6**

**(d) 𝑝 can only take one of three values: 0.2, 0.7, or 0.9.**
* **Prior:** We assume a uniform prior, i.e., each value has equal probability.
* **Posterior:** Calculate the likelihood for each value:
    * $L(0.2) = 45 (0.2)^8 (0.8)^2 \approx 1.15 \times 10^{-5}$
    * $L(0.7) = 45 (0.7)^8 (0.3)^2 \approx 0.0307$
    * $L(0.9) = 45 (0.9)^8 (0.1)^2 \approx 0.0194$
* **Estimated 𝑝:** The value with the highest likelihood is 0.7.
* **(1) Estimated 𝑝 = 0.7**
* **(2) Yes, 𝑝 > 0.6**

**(e) 𝑝 is restricted to the range 0.4 ≤ 𝑝 ≤ 0.9.**
* **Prior:** Uniform distribution over [0.4, 0.9].
* **Posterior:** $P(p|X=8) \propto p^8 (1-p)^2$ within the range [0.4, 0.9].
* **Estimated 𝑝:** From (a), the maximum likelihood occurs at 0.8, which is within the range.
* **(1) Estimated 𝑝 = 0.8**
* **(2) Yes, 𝑝 > 0.6**

**Summary:**

* **(a) 𝑝 = 0.8, Yes**
* **(b) 𝑝 = 0.65, Yes**
* **(c) 𝑝 ≈ 0.888, Yes**
* **(d) 𝑝 = 0.7, Yes**
* **(e) 𝑝 = 0.8, Yes**