my %draw = (map {$_ => 1} qw/ ... /);
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This is a good idea. [Minor point: the parentheses aren't necessary.]

my $hits = @draw{@$bet};
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This is a less good idea: it isn't doing what you appear to think it does. See the following (and the examples that follow it) in "perldata: Slices":

"Slices in scalar context return the last item of the slice."

So, $hits will only ever equal 1 or undef, and the occasional correct output (e.g. "06 14 24 26 42 48 --> 1") will be just coincidence.

Update: Just it case it isn't clear: in list context, @draw{@$bet} will always contain six elements, each of which will be either 1 or undef.