Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine the correct probability distribution for the number of times yellow appears in two spins of the spinner, we need to calculate the probabilities for 0, 1, and 2 appearances of yellow.
The spinner has four sections: red (R), blue (B), green (G), and yellow (Y). The possible outcomes for two spins can be enumerated as follows:
[tex]\[ S = \{RR, RB, RG, RY, BR, BB, BG, BY, GR, GB, GG, GY, YR, YB, YG, YY\} \][/tex]
Next, we calculate the probabilities for each scenario:
1. Probability of 0 yellows (x = 0)
To have 0 yellows means neither spin can result in yellow. Therefore, the outcomes that don't include Y are:
[tex]\[ RR, RB, RG, BR, BB, BG, GR, GB, GG \][/tex]
There are 9 such outcomes. Since each spin is independent and the probability of not landing on yellow for one spin is [tex]\( \frac{3}{4} \)[/tex], the probability for two spins not landing on yellow is:
[tex]\[ P(x=0) = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) = \frac{9}{16} = 0.5625 \][/tex]
2. Probability of 1 yellow (x = 1)
To have exactly 1 yellow means one spin results in yellow and the other does not. The relevant outcomes are:
[tex]\[ RY, BY, GY, YR, YB, YG \][/tex]
There are 6 such outcomes. The probability of one spin resulting in yellow and the other not can be computed as:
[tex]\[ P(x=1) = 2 \times \left(\frac{1}{4}\right) \times \left(\frac{3}{4}\right) = 2 \times \frac{3}{16} = \frac{6}{16} = 0.375 \][/tex]
3. Probability of 2 yellows (x = 2)
To have 2 yellows means both spins result in yellow. The only relevant outcome is:
[tex]\[ YY \][/tex]
There is only 1 such outcome. The probability for both spins resulting in yellow is:
[tex]\[ P(x=2) = \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \frac{1}{16} = 0.0625 \][/tex]
Comparing these calculated probabilities with the given tables, the correct probability distribution is:
\begin{tabular}{|c|c|}
\hline
Yellow: [tex]\( x \)[/tex] & Probability: [tex]\( P_2(x) \)[/tex] \\
\hline
0 & 0.5625 \\
\hline
1 & 0.375 \\
\hline
2 & 0.0625 \\
\hline
\end{tabular}
However, note that the closest matching pre-provided answer table is:
\begin{tabular}{|c|c|}
\hline
Yellow: [tex]\( x \)[/tex] & Prokability: [tex]\( P_2(x) \)[/tex] \\
\hline
0 & 0.5 \\
\hline
1 & 0.375 \\
\hline
2 & 0.125 \\
\hline
\end{tabular}
There might be a typo in the calculated probabilities in the problems provided since the exact matching set isn't present.
The spinner has four sections: red (R), blue (B), green (G), and yellow (Y). The possible outcomes for two spins can be enumerated as follows:
[tex]\[ S = \{RR, RB, RG, RY, BR, BB, BG, BY, GR, GB, GG, GY, YR, YB, YG, YY\} \][/tex]
Next, we calculate the probabilities for each scenario:
1. Probability of 0 yellows (x = 0)
To have 0 yellows means neither spin can result in yellow. Therefore, the outcomes that don't include Y are:
[tex]\[ RR, RB, RG, BR, BB, BG, GR, GB, GG \][/tex]
There are 9 such outcomes. Since each spin is independent and the probability of not landing on yellow for one spin is [tex]\( \frac{3}{4} \)[/tex], the probability for two spins not landing on yellow is:
[tex]\[ P(x=0) = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) = \frac{9}{16} = 0.5625 \][/tex]
2. Probability of 1 yellow (x = 1)
To have exactly 1 yellow means one spin results in yellow and the other does not. The relevant outcomes are:
[tex]\[ RY, BY, GY, YR, YB, YG \][/tex]
There are 6 such outcomes. The probability of one spin resulting in yellow and the other not can be computed as:
[tex]\[ P(x=1) = 2 \times \left(\frac{1}{4}\right) \times \left(\frac{3}{4}\right) = 2 \times \frac{3}{16} = \frac{6}{16} = 0.375 \][/tex]
3. Probability of 2 yellows (x = 2)
To have 2 yellows means both spins result in yellow. The only relevant outcome is:
[tex]\[ YY \][/tex]
There is only 1 such outcome. The probability for both spins resulting in yellow is:
[tex]\[ P(x=2) = \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \frac{1}{16} = 0.0625 \][/tex]
Comparing these calculated probabilities with the given tables, the correct probability distribution is:
\begin{tabular}{|c|c|}
\hline
Yellow: [tex]\( x \)[/tex] & Probability: [tex]\( P_2(x) \)[/tex] \\
\hline
0 & 0.5625 \\
\hline
1 & 0.375 \\
\hline
2 & 0.0625 \\
\hline
\end{tabular}
However, note that the closest matching pre-provided answer table is:
\begin{tabular}{|c|c|}
\hline
Yellow: [tex]\( x \)[/tex] & Prokability: [tex]\( P_2(x) \)[/tex] \\
\hline
0 & 0.5 \\
\hline
1 & 0.375 \\
\hline
2 & 0.125 \\
\hline
\end{tabular}
There might be a typo in the calculated probabilities in the problems provided since the exact matching set isn't present.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
