## Probability of most expensive item automatically being purchased

Hi all, many thanks for any input on this perplexing problem:

If a sporting good store offers a coupon as follows:

Free soccer ball OR free T-shirt OR free Hat

And their POS system automatically credits the highest purchase priced item to the individual if it is in their shopping basket.

So if the
Hat costs \$10
T-Shirt costs \$12
Soccer ball costs \$15

Then the person who purchases a T-shirt AND soccer ball will receive the soccer ball free, while someone who purchases a Hat and T-shirt, would receive the T-shirt free, and so forth.

Let's say the
Hat is included in 5% of all orders
T-shirt is included in 7% of all orders
Soccer ball is included in 10% of all orders

How do we determine the probability of any one of these items being credited to the user at checkout?

My theory is the answer would be:
P(Hat Discounted) = P(Hat) * (1 - P(T-Shirt)) * (1 - P(SB))
(Since either of the more expensive items being added would supercede the hat and cancel its discount)

P(T-Shirt Discounted) = P(T-shirt) * (1 - P(SB))
(Since the presence of the Hat would not impact the T-Shirt hat given the hat is cheaper)

Am I missing something? Should I be subtracting the probability of both other items or something? I think I have the answer, but don't have the privilege of being sure until it's too late.

Many thanks!