Probability - Question

Question

The students in a college need to attend a course and pass three tests. The probability of passing the first test is 0.8 and if a student passes a test, the probability that the student will pass the subsequent test is 0.7. Instead, if the student fails, the probability that the student will fail the subsequent test is 0.6.

Find the probability that the student will pass the first and the third test?

Find the probability that the student will pass at least two tests.

Answer 1

Here is a probability tree for this situation:

.----PPP

|0.7

|

.----PP---|

| |0.3

|0.7 |----PPF

|

.-------P-------.|

| |

| |0.3 .----PFP

| | |0.4

|0.8 | |

| .----PF---|

| |0.6

| |----PFF

|

Start: T1-.|

| .----PPP

| |0.7

| |

|0.2 .----PP---|

| | |0.3

| |0.4 |----PPF

| |

.-------F-------.|

|

|0.6 .----PFP

| |0.4

| |

.----PF---|

|0.6

|----PFF

One outcome of this experiment is one path from the far left to the far right, and the probability of that path is the product of the probabilities on the vertical edges in that path. For example, the probability of PFP (pass test 1, then fail test 2, then pass test 3) is 0.8 x 0.3 x 0.4 while the probability of PPP is 0.8 x 0.7 x 0.7. And those are the two mutually exclusive ways to pass the first and third tests, so the probability of passing the first and third test is (0.8 x 0.3 x 0.4) + (0.8 x 0.7 x 0.7) = 0.096 + 0.392 = 0.488 .