3 mistakes learners make with probability without replacement, each one named and explained.
If something was removed, both the top and the bottom of the fraction change.
The black count drops to 3 but the total stays at 10.
The change to the colour you are tracking is the one you notice.
Question. 4 black, 6 white, one black removed
A common answer. 3/10
The answer. 3/9
Why. There are only nine balls left in the bag altogether.
Removing one ball always changes the total. That part happens whatever colour it was.
The denominator drops to 9 but the black count stays at 4.
Something clearly got smaller, and the total is the more obvious of the two.
Question. 4 black, 6 white, one black removed
A common answer. 4/9
The answer. 3/9
Why. The ball removed was black, so the black count fell too.
Ask what colour was removed. That decides whether the top changes as well as the bottom.
After a black ball is drawn and kept out, the next black is still given as 4/10.
The first probability is the one you just worked out, so reusing it feels efficient.
Question. 4 black, 6 white, one black removed
A common answer. 4/10
The answer. 3/9
Why. There are now three black balls among nine, because one of each count has gone.
If it is not put back, both numbers change. Redraw the bag after every draw.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.