Recognising independent events
Recognising independent events.
- At each split, the outgoing probabilities sum to 1.
- Multiply along a route; add mutually exclusive routes that meet the question.
- Without replacement, update the counts and total for the second draw.
Example 1
A bag contains 2 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
Replacement restores the same composition, so the next probabilities stay unchanged.
Example 2
A bag contains 3 red and 3 blue counters. Two counters are drawn without replacement. Is the second colour independent of the first?
The first draw changes the bag’s composition, so it changes the second-draw probabilities.
Your turn
A bag contains 4 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
Check answer
Replacement restores the same composition, so the next probabilities stay unchanged.
A bag contains 5 red and 3 blue counters. Two counters are drawn without replacement. Is the second colour independent of the first?
Check answer
The first draw changes the bag’s composition, so it changes the second-draw probabilities.
A bag contains 6 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
Check answer
Replacement restores the same composition, so the next probabilities stay unchanged.
A bag contains 7 red and 3 blue counters. Two counters are drawn without replacement. Is the second colour independent of the first?
Check answer
The first draw changes the bag’s composition, so it changes the second-draw probabilities.
A bag contains 8 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
Check answer
Replacement restores the same composition, so the next probabilities stay unchanged.
A bag contains 4 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
AnswerReplacement restores the same composition, so the next probabilities stay unchanged.
A bag contains 5 red and 3 blue counters. Two counters are drawn without replacement. Is the second colour independent of the first?
AnswerThe first draw changes the bag’s composition, so it changes the second-draw probabilities.
A bag contains 6 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
AnswerReplacement restores the same composition, so the next probabilities stay unchanged.
A bag contains 7 red and 3 blue counters. Two counters are drawn without replacement. Is the second colour independent of the first?
AnswerThe first draw changes the bag’s composition, so it changes the second-draw probabilities.
A bag contains 8 red and 3 blue counters. Two counters are drawn with replacement and mixing. Is the second colour independent of the first?
AnswerReplacement restores the same composition, so the next probabilities stay unchanged.