Transcript 12.4 Multiplying Probabilities

Probability of 2 Independent
Events
 If two events, A and B, are independent, then the
probability of both events occurring is
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵)
 Recall: Independent events – current choices do not
affect future choices.
Example – Two Independent Events
 At a picnic, your friend reaches into an ice-filled cooler
containing 8 regular soft drinks and 5 diet drinks. Your
friend removes the can, but then decide that you he is not
really thirsty, and puts it back. What is the probability that
your friend and the next person to reach into the cooler
both randomly select a regular soft drink?
 Event A – The soft drink your friend selects
 Event B – The soft drink the next person selects
𝑃 𝐴 =
𝑃 𝐵 =
 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =
Example – Two Independent Events
 At a picnic, your friend reaches into an ice-filled cooler
containing 8 regular soft drinks and 5 diet drinks. Your
friend removes the can, but then decide that you he is not
really thirsty, and puts it back. What is the probability that
your friend and the next person to reach into the cooler
both randomly select a regular soft drink?
 Event A – The soft drink your friend selects
 Event B – The soft drink the next person selects
𝑃 𝐴 =
8
13
𝑃 𝐵 =
 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =
Example – Two Independent Events
 At a picnic, your friend reaches into an ice-filled cooler
containing 8 regular soft drinks and 5 diet drinks. Your friend
removes the can, but then decide that you he is not really thirsty,
and puts it back. What is the probability that your friend and the
next person to reach into the cooler both randomly select a
regular soft drink?
 Event A – The soft drink your friend selects
 Event B – The soft drink the next person selects
 𝑃 𝐴 =
 𝑃 𝐵 =
8
13
8
13
 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =
Example – Two Independent Events
 At a picnic, your friend reaches into an ice-filled cooler
containing 8 regular soft drinks and 5 diet drinks. Your friend
removes the can, but then decide that you he is not really thirsty,
and puts it back. What is the probability that your friend and the
next person to reach into the cooler both randomly select a
regular soft drink?
 Event A – The soft drink your friend selects
 Event B – The soft drink the next person selects
 𝑃 𝐴 =
 𝑃 𝐵 =
8
13
8
13
 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃 𝐵 =
Example – Two Independent Events
 At a picnic, your friend reaches into an ice-filled cooler
containing 8 regular soft drinks and 5 diet drinks. Your friend
removes the can, but then decide that you he is not really thirsty,
and puts it back. What is the probability that your friend and the
next person to reach into the cooler both randomly select a
regular soft drink?
 Event A – The soft drink your friend selects
 Event B – The soft drink the next person selects
 𝑃 𝐴 =
 𝑃 𝐵 =
8
13
8
13
 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃 𝐵 =
8
13
8
13
=
64
169
Example
 Gerardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both coins he selects are dimes?
 A – First coin is a dime
 B- Second coin is a dime
Example
 Gerardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both coins he selects are dimes?
 A – First coin is a dime
 B- Second coin is a dime
9
P ( A) 
16
Example
 Gerardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both coins he selects are dimes?
 A – First coin is a dime
 B- Second coin is a dime
9
P ( A) 
16
9
P( B) 
16
Example
 Gerardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both coins he selects are dimes?
 A – First coin is a dime
 B- Second coin is a dime
9
P ( A) 
16
9
P( B) 
16
 9  9  81
P( AandB)     
 16  16  256
Example – 3 Independent Events
 When three dice are rolled, what is the probability that
2 dice show a 5 and the third dice shows an even
number?
 Event A – First die is a 5
 Event B – Second die is a 5
 Event C – Third die is even
Example – 3 Independent Events
 When three dice are rolled, what is the probability that
2 dice show a 5 and the third dice shows an even
number?
 Event A – First die is a 5
 Event B – Second die is a 5
 Event C – Third die is even
1
P( A) 
6
Example – 3 Independent Events
 When three dice are rolled, what is the probability that
2 dice show a 5 and the third dice shows an even
number?
 Event A – First die is a 5
 Event B – Second die is a 5
 Event C – Third die is even
1
P( A) 
6
1
P( B) 
6
Example – 3 Independent Events
 When three dice are rolled, what is the probability that
2 dice show a 5 and the third dice shows an even
number?
 Event A – First die is a 5
 Event B – Second die is a 5
 Event C – Third die is even
1
P( A) 
6
1
P( B) 
6
1
P (C ) 
2
Example – 3 Independent Events
 When three dice are rolled, what is the probability that
2 dice show a 5 and the third dice shows an even
number?
1
P( A) 
6
1
P( B) 
 Event A – First die is a 5
6
 Event B – Second die is a 5
1
 Event C – Third die is even
P (C ) 
2
 1  1  1  1
P( A  B  C )      
 6  6  2  72
Symbol for AND
Assignment