document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161858>Question 1161858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the probability that a passenger will miss a flight is 0.0957. Airlines do not like flights with empty​ seats, but it is also not desirable to have overbooked flights because passengers must be​ \"bumped\" from the flight. Suppose that an airplane has a seating capacity of 59 passengers.<BR>\n" );
document.write( "​(a) If 61 tickets are​ sold, what is the probability that 60 or 61 passengers show up for the flight resulting in an overbooked​ flight?<BR>\n" );
document.write( "​(b) Suppose that 65 tickets are sold. What is the probability that a passenger will have to be​ \"bumped\"?<BR>\n" );
document.write( "​(c) For a plane with seating capacity of<BR>\n" );
document.write( "52 ​passengers, how many tickets may be sold to keep the probability of a passenger being​ \"bumped\" below 5​%? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854411 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854411\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "from scipy.stats import binom\r<BR>\n" );
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document.write( "# Constants<BR>\n" );
document.write( "p_miss = 0.0957<BR>\n" );
document.write( "p_show = 1 - p_miss\r<BR>\n" );
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document.write( "# (a) n=61, capacity=59, find P(k=60) + P(k=61)<BR>\n" );
document.write( "n_a = 61<BR>\n" );
document.write( "p_60 = binom.pmf(60, n_a, p_show)<BR>\n" );
document.write( "p_61 = binom.pmf(61, n_a, p_show)<BR>\n" );
document.write( "ans_a = p_60 + p_61\r<BR>\n" );
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document.write( "# (b) n=65, capacity=59, find P(k > 59)<BR>\n" );
document.write( "n_b = 65<BR>\n" );
document.write( "capacity_b = 59<BR>\n" );
document.write( "ans_b = binom.sf(capacity_b, n_b, p_show) # sf is 1 - cdf, so P(X > 59)\r<BR>\n" );
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document.write( "# (c) capacity=52, find max n such that P(k > 52) < 0.05<BR>\n" );
document.write( "capacity_c = 52<BR>\n" );
document.write( "threshold = 0.05<BR>\n" );
document.write( "best_n = 52<BR>\n" );
document.write( "for n in range(53, 100):<BR>\n" );
document.write( "    prob_bumped = binom.sf(capacity_c, n, p_show)<BR>\n" );
document.write( "    if prob_bumped < threshold:<BR>\n" );
document.write( "        best_n = n<BR>\n" );
document.write( "    else:<BR>\n" );
document.write( "        break\r<BR>\n" );
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document.write( "print(f\"Part (a): {ans_a=}\")<BR>\n" );
document.write( "print(f\"Part (b): {ans_b=}\")<BR>\n" );
document.write( "print(f\"Part (c): {best_n=}, {binom.sf(capacity_c, best_n, p_show)=}, {binom.sf(capacity_c, best_n+1, p_show)=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Part (a): ans_a=0.016126517739814418<BR>\n" );
document.write( "Part (b): ans_b=0.4014073805745022<BR>\n" );
document.write( "Part (c): best_n=54, binom.sf(capacity_c, best_n, p_show)=0.029369939359098, binom.sf(capacity_c, best_n+1, p_show)=0.09276094534760537\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To solve these problems, we use the **Binomial Distribution**. <BR>\n" );
document.write( "* Let $n$ be the number of tickets sold.<BR>\n" );
document.write( "* Let $p$ be the probability that a passenger shows up for the flight. Given the probability of missing is $0.0957$, the probability of showing up is $p = 1 - 0.0957 = 0.9043$.<BR>\n" );
document.write( "* Let $X$ be the number of passengers who show up. $X \sim \text{Binomial}(n, 0.9043)$.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
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document.write( "### (a) If 61 tickets are sold, what is the probability that 60 or 61 passengers show up?<BR>\n" );
document.write( "We need to calculate $P(X=60) + P(X=61)$ for $n=61$.\r<BR>\n" );
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document.write( "Using the binomial formula $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$:<BR>\n" );
document.write( "* $P(X=60) = \binom{61}{60} (0.9043)^{60} (0.0957)^1 \approx 0.0139$<BR>\n" );
document.write( "* $P(X=61) = \binom{61}{61} (0.9043)^{61} (0.0957)^0 \approx 0.0022$\r<BR>\n" );
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document.write( "$$P(60 \text{ or } 61) = 0.01392 + 0.00221 \approx 0.0161$$\r<BR>\n" );
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document.write( "**Probability: 0.0161**\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### (b) If 65 tickets are sold, what is the probability that a passenger will be \"bumped\"?<BR>\n" );
document.write( "A passenger is bumped if more than 59 people show up ($X > 59$). For $n=65$:<BR>\n" );
document.write( "$$P(X > 59) = P(X=60) + P(X=61) + P(X=62) + P(X=63) + P(X=64) + P(X=65)$$<BR>\n" );
document.write( "Using the cumulative distribution function (or summing the individual probabilities):<BR>\n" );
document.write( "$$P(X > 59) \approx 0.4014$$\r<BR>\n" );
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document.write( "**Probability: 0.4014**\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
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document.write( "### (c) For a capacity of 52, how many tickets may be sold to keep the probability of being \"bumped\" below 5%?<BR>\n" );
document.write( "We need to find the largest $n$ such that $P(X > 52) < 0.05$.\r<BR>\n" );
document.write( "\n" );
document.write( "* If **$n=53$** tickets are sold: $P(X > 52) = P(X=53) = (0.9043)^{53} \approx 0.0050$. (This is below 5%)<BR>\n" );
document.write( "* If **$n=54$** tickets are sold: $P(X > 52) = P(X=53) + P(X=54) \approx 0.0294$. (This is below 5%)<BR>\n" );
document.write( "* If **$n=55$** tickets are sold: $P(X > 52) \approx 0.0928$. (This exceeds 5%)\r<BR>\n" );
document.write( "\n" );
document.write( "**Maximum tickets that can be sold: 54** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );