The probability of Mary, Esther and John coming to school late on Monday are 1/4,2/5 and 1/3 respectively
(a).draw a tree diagram to represent the information
(b). calculate the probability that
(1).all the three girls are late
(11).all except Esther are late
(111).at most two girls are late
To answer these questions, we can use a tree diagram to represent the different possibilities. Here's how you can create the tree diagram:
(a) Drawing the Tree Diagram:
1. Start by drawing a vertical line from the top to the bottom of your paper.
2. At the top of the line, write "Monday" to indicate the day we are considering.
3. Draw three branches coming out from the line, representing Mary, Esther, and John.
4. On the Mary branch, write "1/4", indicating the probability of her being late.
5. On the Esther branch, write "2/5", indicating the probability of her being late.
6. On the John branch, write "1/3", indicating the probability of him being late.
The tree diagram should now look like this:
Monday
/ | \
1/4 2/5 1/3
Mary Esther John
Now, let's move on to calculating the probabilities:
(b) Calculating the Probabilities:
(1) Probability that all three girls are late:
To find this probability, you will need to multiply the probabilities of each girl being late. So, multiply 1/4 (Mary's probability) by 2/5 (Esther's probability) by 2/3 (John's probability, since he is not a girl).
Calculating: (1/4) * (2/5) * (2/3) = 4/60 = 1/15. Therefore, the probability that all three girls are late is 1/15.
(11) Probability that all except Esther are late:
This means we need Mary and John to be late, but Esther to be on time. To find this probability, multiply Mary's probability of being late (1/4) by John's probability of being late (1/3) by Esther's probability of being on time (1 - 2/5, since Esther being late is the complement of her being on time).
Calculating: (1/4) * (2/3) * (1 - 2/5) = (1/4) * (2/3) * (3/5) = 6/60 = 1/10. Therefore, the probability that all except Esther are late is 1/10.
(111) Probability that at most two girls are late:
This means either none of the girls are late, one girl is late, or two girls are late. To find this probability, you can find the probability of each scenario and add them up.
- None of the girls are late: Multiply Mary's probability of being on time (1 - 1/4) by Esther's probability of being on time (1 - 2/5) by John's probability of being on time (1 - 1/3).
Calculating: (1 - 1/4) * (1 - 2/5) * (1 - 1/3) = (3/4) * (3/5) * (2/3) = 18/60 = 3/10.
- One girl is late: This can happen in three ways (Mary late, Esther late, or John late). Calculate each possibility separately and add them up.
- Mary late: Multiply Mary's probability of being late (1/4, Mary) by Esther's probability of being on time (1 - 2/5) by John's probability of being on time (1 - 1/3).
Calculating: (1/4) * (1 - 2/5) * (1 - 1/3) = (1/4) * (3/5) * (2/3) = 6/60 = 1/10.
- Esther late: Multiply Mary's probability of being on time (1 - 1/4) by Esther's probability of being late (2/5, Esther) by John's probability of being on time (1 - 1/3).
Calculating: (1 - 1/4) * (2/5) * (1 - 1/3) = (3/4) * (2/5) * (2/3) = 12/60 = 1/5.
- John late: Multiply Mary's probability of being on time (1 - 1/4) by Esther's probability of being on time (1 - 2/5) by John's probability of being late (1/3, John).
Calculating: (1 - 1/4) * (1 - 2/5) * (1/3) = (3/4) * (3/5) * (1/3) = 9/60 = 3/20.
Adding up these three possibilities: (1/10) + (1/5) + (3/20) = 10/60 + 12/60 + 9/60 = 31/60.
Therefore, the probability that at most two girls are late is 31/60.
I hope this explanation helps! Let me know if you have any further questions.