document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200098>Question 1200098</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 10. Suppose it is known that the probability of recovery for a certain disease is 0.4. If 35 people are stricken with the disease, what is the probability that:<BR>\n" );
document.write( "(a) 25 or more will recover?<BR>\n" );
document.write( "(b) Fewer than five will recover?<BR>\n" );
document.write( "(Use the normal approximation.)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848095 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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=848095\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the parameters:**\r<BR>\n" );
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document.write( "* **p:** Probability of recovery = 0.4<BR>\n" );
document.write( "* **n:** Number of people stricken with the disease = 35\r<BR>\n" );
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document.write( "**2. Calculate the mean and standard deviation:**\r<BR>\n" );
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document.write( "* **Mean (μ):** μ = n * p = 35 * 0.4 = 14<BR>\n" );
document.write( "* **Standard deviation (σ):** σ = √(n * p * (1 - p)) = √(35 * 0.4 * 0.6) ≈ 2.898\r<BR>\n" );
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document.write( "**3. (a) Probability of 25 or more recovering:**\r<BR>\n" );
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document.write( "* **Continuity correction:** Since we're using the normal approximation to the binomial distribution, we need to apply a continuity correction. For \"25 or more,\" we'll use 24.5 as the lower bound.\r<BR>\n" );
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document.write( "* **Calculate the z-score:**<BR>\n" );
document.write( "   z = (X - μ) / σ = (24.5 - 14) / 2.898 ≈ 3.62\r<BR>\n" );
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document.write( "* **Find the probability:** <BR>\n" );
document.write( "   Using a standard normal distribution table or calculator, we find the area to the right of z = 3.62. <BR>\n" );
document.write( "   P(X ≥ 25) ≈ 0.00015\r<BR>\n" );
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document.write( "**4. (b) Probability of fewer than 5 recovering:**\r<BR>\n" );
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document.write( "* **Continuity correction:** For \"fewer than 5,\" we'll use 4.5 as the upper bound.\r<BR>\n" );
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document.write( "* **Calculate the z-score:**<BR>\n" );
document.write( "   z = (X - μ) / σ = (4.5 - 14) / 2.898 ≈ -3.28\r<BR>\n" );
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document.write( "* **Find the probability:**<BR>\n" );
document.write( "   Using a standard normal distribution table or calculator, we find the area to the left of z = -3.28. <BR>\n" );
document.write( "   P(X < 5) ≈ 0.0005\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* (a) The probability that 25 or more people will recover is approximately **0.00015 (or 0.015%)**.<BR>\n" );
document.write( "* (b) The probability that fewer than 5 people will recover is approximately **0.0005 (or 0.05%)**.\r<BR>\n" );
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document.write( "**Note:** These probabilities are based on the normal approximation to the binomial distribution. For a more precise calculation, you could use the binomial probability formula directly.<BR>\n" );
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