A company experiments with three website designs to see which one would lead to the highest probability of purchase. The first$($design A$)$ used enhanced product$$ information, the second$($design B$)$ used extensive$$ iconography, and the third$($design C$)$ allowed the customer to submit their own product ratings. After $6$ weeks of$$ testing, the designs delivered probabilities of purchase of $9\%,6\%,$ and $3\%$ respectively. Equal numbers of customers were sent randomly to each website design. Complete parts a through d below.

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Part $1$

$$a$)$ Construct a probability tree for this experiment.

Purchase

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Design A and Purchase

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Design A

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No Purchase

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Design A and No Purchase

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Purchase

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Design B and Purchase

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Design B

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No Purchase

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Design B and No Purchase

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Purchase

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Design C and Purchase

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Design C

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No Purchase

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Design C and No Purchase

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$($Round to three decimal places as$$ needed.$)$