document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166976>Question 1166976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A special deck of cards has 10 green cards , 13 blue cards , and 5 red cards . When a card is picked, the color is recorded. An experiment consists of first picking a card and then tossing a coin.\r<BR>\n" );
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document.write( "a. How many elements are there in the sample space? \r<BR>\n" );
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document.write( "b. Let A be the event that a green card is picked first, followed by landing a head on the coin toss.<BR>\n" );
document.write( "    P(A) = <BR>\n" );
document.write( " Round your answer to 4 decimal places.\r<BR>\n" );
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document.write( "c. Let B be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive?\r<BR>\n" );
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document.write( "- No, they are not Mutually Exclusive<BR>\n" );
document.write( "-Yes, they are Mutually Exclusive\r<BR>\n" );
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document.write( "d. Let C be the event that a green or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive?\r<BR>\n" );
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document.write( " -No, they are not Mutually Exclusive<BR>\n" );
document.write( "-Yes, they are Mutually Exclusive<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852344 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=852344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the solutions for the experiment involving picking a card and then tossing a coin.\r<BR>\n" );
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document.write( "## a. How many elements are there in the sample space?\r<BR>\n" );
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document.write( "The experiment consists of two independent steps:<BR>\n" );
document.write( "1.  **Picking a card:** There are three possible outcomes (Green, Blue, Red).<BR>\n" );
document.write( "    * Total cards: $10 + 13 + 5 = 28$.<BR>\n" );
document.write( "2.  **Tossing a coin:** There are two possible outcomes (Head, Tail).\r<BR>\n" );
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document.write( "The total number of elements in the sample space ($S$) is the product of the outcomes of each step:<BR>\n" );
document.write( "$$\text{Elements in } S = (\text{Card Outcomes}) \times (\text{Coin Outcomes}) = 3 \times 2 = \mathbf{6}$$\r<BR>\n" );
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document.write( "The actual sample space is: $\{ (G, H), (G, T), (B, H), (B, T), (R, H), (R, T) \}$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## b. Let A be the event that a green card is picked first, followed by landing a head on the coin toss.\r<BR>\n" );
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document.write( "The event $A$ is the outcome $(G, H)$. Since the two events are independent, $P(A) = P(\text{Green}) \times P(\text{Head})$.\r<BR>\n" );
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document.write( "1.  **Probability of Green Card:**<BR>\n" );
document.write( "    $$P(G) = \frac{\text{Number of Green Cards}}{\text{Total Cards}} = \frac{10}{28}$$<BR>\n" );
document.write( "2.  **Probability of Head:**<BR>\n" );
document.write( "    $$P(H) = \frac{1}{2} = 0.5$$\r<BR>\n" );
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document.write( "3.  **Probability of Event A:**<BR>\n" );
document.write( "    $$P(A) = P(G) \times P(H) = \frac{10}{28} \times 0.5 = \frac{5}{28}$$<BR>\n" );
document.write( "    $$P(A) \approx 0.178571...$$\r<BR>\n" );
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document.write( "Rounding to 4 decimal places:<BR>\n" );
document.write( "$$P(A) = \mathbf{0.1786}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## c. Let B be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive?\r<BR>\n" );
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document.write( "* Event A: A green card is picked, followed by a Head. $A = \{(G, H)\}$<BR>\n" );
document.write( "* Event B: A red or blue card is picked, followed by a Head. $B = \{(R, H), (B, H)\}$\r<BR>\n" );
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document.write( "Two events are **mutually exclusive** if they cannot occur at the same time, meaning their intersection is empty ($A \cap B = \emptyset$).\r<BR>\n" );
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document.write( "Since the card pick must be either Green (for A) or Red/Blue (for B), there are no outcomes they share.\r<BR>\n" );
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document.write( "**Answer:** **Yes, they are Mutually Exclusive**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## d. Let C be the event that a green or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive?\r<BR>\n" );
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document.write( "* Event A: A green card is picked, followed by a Head. $A = \{(G, H)\}$<BR>\n" );
document.write( "* Event C: A green or blue card is picked, followed by a Head. $C = \{(G, H), (B, H)\}$\r<BR>\n" );
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document.write( "The intersection of $A$ and $C$ is the outcome $(G, H)$. Since $A \cap C = \{(G, H)\}$, the intersection is not empty, and $P(A \cap C) > 0$.\r<BR>\n" );
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document.write( "**Answer:** **No, they are not Mutually Exclusive** </SPAN>\n" );
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