Representing Data for Finding Probabilities There are 35 students 20 take math 25 take science 15 take both Venn Diagram Contingency table M^M S ^S

Conditional Probability P(B|A) means the probability B happens, given that A happens P(Science|Math) = 15/20 P(Math|Science) = 15/25 You can get this from either representation M^M S ^S

Independence Two events, A and B are independent if P(B|A)=P(B) Example: I roll a die and flip a coin. P(H|6)=P(H) because the number on the die does not affect my chance of getting heads. Heads and Getting a Six are independent events.

Independence Two events, A and B are independent if P(B|A)=P(B) Example: I pick a card out of a regular 52 card deck. Then, I pick a second card without replacing the first. Are the events getting a red card and then getting a queen independent? No, because removing a card from the deck changes my probabilities for the second draw. My denominator is now 51 instead of 52.

Independence Two events, A and B are independent if P(B|A)=P(B) Example: I pick a card out of a regular 52 card deck. Then, I pick a second card after putting the first card back in the deck. Are the events getting a red card and then getting a queen independent? Yes, because replacing the first card makes it as if I never drew it in the first place. All probabilities remain the same for the second draw

Multiplication Rule P(A and B) = P(A)*P(B|A) If A and B are independent, this becomes: P(A and B) = P(A)*P(B)

Multiplication Rule Experiment: Flip a Coin and Roll a Die. H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 HTHT What is the probability of getting a head and then an even number? P(H and E) =P(H)*P(E) = P(H and E) =

Multiplication Rule There are 3 blue marbles and 2 green in a box. What is the probability you draw a blue and then a green? P(B) = 3/5 P(G|B) = 1/2 X

Multiplication Rule There are 3 blue marbles and 2 green in a box. B G 3/5 2/5 BGBGBGBG 2/4 1/4 3/4 3/10 1/10 P(B and G) = 3/10 P(G and G) = 1/10 P(G|B) = 2/4=1/2 P(G|G) = 1/4

Multiplication Rule What is the probability that I draw a queen and then a Ten when drawing without replacement? P(Q) = 4/52 P(Ten|Q) = 4/51 P(Q and ten) = What is the probability that I draw a queen and then a Ten when drawing with replacement? P(Q) = 4/52 P(Ten|Q) = 4/52 P(Q and ten) =

Multiplication Rule Here are the results for 240 students taking an entrance exam for placement in upper mathematics at a high school: 80 Passed Female Is a student’s passing status independent of gender? If so, P(passed) =P(passed|female) P(passed) =160/240 = 2/3 P(passed|female)=80/120=2/3 P(passed)=P(passed|female)  Passing is independent of gender for these results