Questions: This tree diagram represents the sample space for all possible results from randomly selecting 2 items from a binomial situation. Use the displayed probabilities of each possible outcome to find each of the probabilities. start 1st item 2nd item (end) result - - - - 0.4 yes a yes 0.6 no b 0.4 yes c 0.6 no d a. Prob(yes and yes) = b. Prob(yes and no in that order) = 0.24 c. Prob(no and yes in that order) = 0.24 d. Prob(no and no) =

This tree diagram represents the sample space for all possible results from randomly selecting 2 items from a binomial situation. Use the displayed probabilities of each possible outcome to find each of the probabilities. start 1st item 2nd item (end) result - - - - 0.4 yes a yes 0.6 no b 0.4 yes c 0.6 no d a. Prob(yes and yes) = b. Prob(yes and no in that order) = 0.24 c. Prob(no and yes in that order) = 0.24 d. Prob(no and no) =
Transcript text: This tree diagram represents the sample space for all possible results from randomly selecting 2 items from a binomial situation. Use the displayed probabilities of each possible outcome to find each of the probabilities. \begin{tabular}{|c|c|c|c|} \hline start & 1st item & 2nd item (end) & result \\ \hline & & 0.4 yes & a \\ \hline & yes & 0.6 no & b \\ \hline & & 0.4 yes & c \\ \hline & & 0.6 no & d \\ \hline \end{tabular} a. $\operatorname{Prob}($ yes and yes $)=$ $\square$ b. Prob(yes and no in that order) $=0.24$ c. $\operatorname{Prob(no}$ and yes in that order) $=0.24$ d. $\operatorname{Prob}($ no and no$)=$ $\square$
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Solution

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Solution Steps

Step 1: Calculate Probability of "Yes and Yes"

To find the probability of selecting "yes" for both items, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

  • \( n = 2 \) (the number of trials),
  • \( x = 2 \) (the number of successes),
  • \( p = 0.4 \) (the probability of success),
  • \( q = 0.6 \) (the probability of failure).

Substituting the values, we find:

\[ P(\text{yes and yes}) = 0.16 \]

Step 2: Calculate Probability of "No and No"

Next, we calculate the probability of selecting "no" for both items using the same formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

  • \( n = 2 \),
  • \( x = 0 \) (the number of successes, which means both are failures),
  • \( p = 0.4 \),
  • \( q = 0.6 \).

Substituting the values, we find:

\[ P(\text{no and no}) = 0.36 \]

Step 3: Given Probabilities

The probabilities for the other outcomes are provided as follows:

  • Probability of "yes and no" in that order:

\[ P(\text{yes and no}) = 0.24 \]

  • Probability of "no and yes" in that order:

\[ P(\text{no and yes}) = 0.24 \]

Final Answer

The calculated probabilities are:

  • \( P(\text{yes and yes}) = 0.16 \)
  • \( P(\text{no and no}) = 0.36 \)
  • \( P(\text{yes and no}) = 0.24 \)
  • \( P(\text{no and yes}) = 0.24 \)

Thus, the final answers are:

\[ \boxed{P(\text{yes and yes}) = 0.16} \] \[ \boxed{P(\text{no and no}) = 0.36} \]

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