{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2.4-2.5 - 2.4 Conditional Probability and Independence...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
2.4 Conditional Probability and Independence Conditional Probability Independent Events Parallel and Serial Systems Bayes’ Theorem Conditional Probability Let A, B two events. The conditional probability of B given condition A (given that A has occurred) is defined to be ) ( ) ( ) | ( A P B and A P A B P = or Example 1 A box contains 4 red and 2 green balls. Draw successively two balls without replacement and observe the color. Denote: G 1 = green on the first draw, G 2 = green on the second draw R 1 = red on the first draw, R 2 = red on the second draw 1 ) ( ) | ( ) ( ) ( ) | ( ) ( A P A B P B and A P B P B A P B and A P = =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Sample space S = { G 1 G 2 , G 1 R 2, R 1 G 2 , R 1 R 2 } Probabilities P(G 1 G 2 ) = P(G 1 ) P(G 2 | G 1 ) = 2/6 ×1/5 = 2/30 P(R 1 R 2 ) = P(R 1 ) P(R 2 | R 1 ) = 4/6 ×3/5 =12/30 P(G 1 R 2 ) = P(G 1 ) P(R 2 | G 1 ) = 2/6 ×4/5 =8/30 P(R 1 G 2 ) = P(R 1 ) P(G 2 | R 1 ) = 4/6 ×2/5 =8/30 The probability that the second ball will be green. P(G 2 ) = P(R 1 G 2 ) + P(G 1 G 2 ) = 8/30+2/30 =10/30. Tree diagram G 2 R 2 G 1 R 1 2 G G R G R R (1) (2)
Background image of page 2
Suppose that green was observed in the second draw. Find the conditional probability that the first ball was also green. = = ) G ( ) G G ( ) G | G ( 2 2 1 2 1 P P P 2 . 0 ) 30 / 8 ( ) 30 / 2 ( 30 / 2 = + , since ). ( ) ( ) ( 2 1 2 1 2 G G P G R P G P + = Finally, compute the probability that exactly one ball selected is green, and probability that at least one ball selected is green (Do It Yourself!) Example 2. Given is a contingency table of 100 students cross- classified by their school goal and gender Goals Gender Grades Popular Sports Total Boy 24 10 13 47 Girl 27 19 7 53 Tota l 51 29 20 100 A student is selected at random. Let G ="a girl is selected" and S = "wants to excel at sports" 1.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}