document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209944>Question 1209944</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coin has a probability 𝑝 of landing heads. It is flipped 10 times, resulting in 8 heads. Using Bayesian inference with the following prior information<BR>\n" );
document.write( "(a) No prior knowledge about 𝑝.<BR>\n" );
document.write( "(b) The coin is likely fair (𝑝 around 0.5).<BR>\n" );
document.write( "(c) The coin is likely biased (𝑝 around 0 or 1).<BR>\n" );
document.write( "(d) 𝑝 can only take one of three values: 0.2, 0.7, or 0.9.<BR>\n" );
document.write( "(e) 𝑝 is restricted to the range 0.4 ≤ 𝑝 ≤ 0.9.<BR>\n" );
document.write( ", answer the following questions:<BR>\n" );
document.write( "(1) What is the estimated value of 𝑝?<BR>\n" );
document.write( "(2) Is 𝑝 greater than 0.6?<BR>\n" );
document.write( "The solution provided by @CPhill is completely incorrect. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851148 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851148\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each case using Bayesian inference:\r<BR>\n" );
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document.write( "**Likelihood:**<BR>\n" );
document.write( "The likelihood of observing 8 heads in 10 flips is given by the binomial distribution:<BR>\n" );
document.write( "$$L(p|X=8) = \binom{10}{8} p^8 (1-p)^2 = 45 p^8 (1-p)^2$$\r<BR>\n" );
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document.write( "**(a) No prior knowledge about 𝑝.**<BR>\n" );
document.write( "* **Prior:** Uniform distribution over [0, 1], i.e., $P(p) \propto 1$.<BR>\n" );
document.write( "* **Posterior:** $P(p|X=8) \propto L(p|X=8) \times P(p) \propto p^8 (1-p)^2$.<BR>\n" );
document.write( "* **Estimated 𝑝:** To find the maximum, we differentiate and set to zero:<BR>\n" );
document.write( "    * $\frac{d}{dp} [p^8 (1-p)^2] = 0$<BR>\n" );
document.write( "    * This leads to $p = 8/10 = 0.8$.<BR>\n" );
document.write( "* **(1) Estimated 𝑝 = 0.8**<BR>\n" );
document.write( "* **(2) Yes, 𝑝 > 0.6**\r<BR>\n" );
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document.write( "**(b) The coin is likely fair (𝑝 around 0.5).**<BR>\n" );
document.write( "* **Prior:** Beta distribution centered around 0.5, e.g., Beta(5, 5), so $P(p) \propto p^4(1-p)^4$.<BR>\n" );
document.write( "    * For the sake of simplicity I will use Beta(6,6), which is also centered at 0.5.<BR>\n" );
document.write( "* **Posterior:** $P(p|X=8) \propto p^8 (1-p)^2 \times p^5 (1-p)^5 = p^{13} (1-p)^7$.<BR>\n" );
document.write( "* **Estimated 𝑝:** Differentiate and set to zero:<BR>\n" );
document.write( "    * $\frac{d}{dp} [p^{13} (1-p)^7] = 0$<BR>\n" );
document.write( "    * This leads to $p = 13/20 = 0.65$.<BR>\n" );
document.write( "* **(1) Estimated 𝑝 = 0.65**<BR>\n" );
document.write( "* **(2) Yes, 𝑝 > 0.6**\r<BR>\n" );
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document.write( "**(c) The coin is likely biased (𝑝 around 0 or 1).**<BR>\n" );
document.write( "* **Prior:** Beta distribution biased towards 0 or 1, e.g., Beta(9, 1). So $P(p) \propto p^8(1-p)^0$.<BR>\n" );
document.write( "* **Posterior** $P(p|X=8) \propto p^8(1-p)^2 * p^8 = p^{16}(1-p)^2$.<BR>\n" );
document.write( "* **Estimated p**<BR>\n" );
document.write( "    * $\frac{d}{dp}(p^{16}(1-p)^2) = 16p^{15}(1-p)^2-2p^{16}(1-p) = 0$<BR>\n" );
document.write( "    * $16(1-p)-2p = 0$<BR>\n" );
document.write( "    * $16-18p = 0$<BR>\n" );
document.write( "    * $p= 16/18 = 8/9 \approx 0.888$.<BR>\n" );
document.write( "* **(1) Estimated 𝑝 ≈ 0.888**<BR>\n" );
document.write( "* **(2) Yes, 𝑝 > 0.6**\r<BR>\n" );
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document.write( "**(d) 𝑝 can only take one of three values: 0.2, 0.7, or 0.9.**<BR>\n" );
document.write( "* **Prior:** We assume a uniform prior, i.e., each value has equal probability.<BR>\n" );
document.write( "* **Posterior:** Calculate the likelihood for each value:<BR>\n" );
document.write( "    * $L(0.2) = 45 (0.2)^8 (0.8)^2 \approx 1.15 \times 10^{-5}$<BR>\n" );
document.write( "    * $L(0.7) = 45 (0.7)^8 (0.3)^2 \approx 0.0307$<BR>\n" );
document.write( "    * $L(0.9) = 45 (0.9)^8 (0.1)^2 \approx 0.0194$<BR>\n" );
document.write( "* **Estimated 𝑝:** The value with the highest likelihood is 0.7.<BR>\n" );
document.write( "* **(1) Estimated 𝑝 = 0.7**<BR>\n" );
document.write( "* **(2) Yes, 𝑝 > 0.6**\r<BR>\n" );
document.write( "\n" );
document.write( "**(e) 𝑝 is restricted to the range 0.4 ≤ 𝑝 ≤ 0.9.**<BR>\n" );
document.write( "* **Prior:** Uniform distribution over [0.4, 0.9].<BR>\n" );
document.write( "* **Posterior:** $P(p|X=8) \propto p^8 (1-p)^2$ within the range [0.4, 0.9].<BR>\n" );
document.write( "* **Estimated 𝑝:** From (a), the maximum likelihood occurs at 0.8, which is within the range.<BR>\n" );
document.write( "* **(1) Estimated 𝑝 = 0.8**<BR>\n" );
document.write( "* **(2) Yes, 𝑝 > 0.6**\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* **(a) 𝑝 = 0.8, Yes**<BR>\n" );
document.write( "* **(b) 𝑝 = 0.65, Yes**<BR>\n" );
document.write( "* **(c) 𝑝 ≈ 0.888, Yes**<BR>\n" );
document.write( "* **(d) 𝑝 = 0.7, Yes**<BR>\n" );
document.write( "* **(e) 𝑝 = 0.8, Yes**<BR>\n" );
document.write( " </SPAN>\n" );
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