The chart below lists the odds that a horse will place in

"The chart below lists the odds that a horse will place in the top three. As part of a contest, if at least 2 of the horses that a person bets upon finish in the top three, the bettor receives a T-shirt. If Cecilia bets on The Baron, Happy Cynic, and Inamorata, what is the probability that she does not receive a T-shirt?"

Cecilia can win a T-Shirt, if at least 2 of The Baron, Happy Cynic, and Inamorata finish in top three.

Now theoretically, when she will not get a T-Shirt ? When either only 1 or NONE of her horses can make into top three.
But practically, there are total 5 horses, so anyhow 1 of her horse will make into top 3, so the NONE case is eliminated.

The winning probability of the 3 horses Cecilia has bet:

The Barron: 3/5
Happy Cynic: 7/10
Inamorata: 1/2

For any game with no draws: P(Win) + P(Loss) = 1

So probability that Cecilia doesn't win T-Shirt = Probability that only one of the above three horses win and other two loose

= P(The Barron Wins)*P(Happy Cynic Looses)*P(Inamorata Looses)
+ P(The Barron Looses)*P(Happy Cynic Wins)*P(Inamorata Looses)
+P(The Barron Looses)*P(Happy Cynic Looses)*P(Inamorata Wins)

= (3/5) * (1-7/10)*(1-1/2) + (1-3/5) * (7/10)*(1-1/2) + (1-3/5) * (1-7/10)*(1/2)

= 29/100