The odds of drawing a particular card in a 60-card deck are obviously 1/60. If there are four such cards, the odds are 4/60. The odds of NOT drawing one of those cards in the first draw is 1 - 4/60 = 56/60.
To calculate the odds of the entire first hand, we can do it backwards:
The odds of not having any of the four cards in the first card is 56/60 (as I said above). The second card has odds of 55/59 (i.e. one of the remaining non-Bird cards after a non-Bird card was drawn to start), and then 54/58 and so on:
- Card 1: 56/60 chance of not being the card you targeted
- Card 2: 55/59
- Card 3: 54/58
- Card 4: 53/57
- Card 5: 52/56
- Card 6: 51/55
- Card 7: 50/54
The odds of ALL of these happening (i.e. none of the four cards being in your hand) is the result of multiplying all these odds together:
(56*55*54*53*52*51*50) / (60*59*58*57*56*55*54) = ~0.6005 or ~60%
To calculate the odds of at least one of these cards being the one you're looking for, you can subtract this result from 1 (or 100%) to get a ~40% chance that (at least) one of your four cards will occur in a 7-card draw from a 60-card deck.