3.3-The Addition Rule

• Mutually Exclusive Events: can NOT occur at the same time

A B A BA and B

A and B areMutually exclusive A and B are NOT

Mutually exclusive

Are these events mutually exclusive?

• 1. A= roll a 3 on a die B = roll a 4 on a die.

• 2. A= roll a 3 on a die B = roll an even # on die

• 3. Select a student A=male B=nursing major

• 4. Select blood donor A=type O B=female

Are these events mutually exclusive?

• 1. A= roll a 3 on a die B = roll a 4 on a die.Yes – can’t occur at same time

• 2. A= roll a 3 on a die B = roll an even # on die

• 3. Select a student A=male B=nursing major

• 4. Select blood donor A=type O B=female

Are these events mutually exclusive?

• 1. A= roll a 3 on a die B = roll a 4 on a die.Yes – can’t occur at same time

• 2. A= roll a 3 on a die B = roll an even # on dieYes – can’t occur at same time

• 3. Select a student A=male B=nursing major

• 4. Select blood donor A=type O B=female

Are these events mutually exclusive?

• 1. A= roll a 3 on a die B = roll a 4 on a die.Yes – can’t occur at same time

• 2. A= roll a 3 on a die B = roll an even # on dieYes – can’t occur at same time

• 3. Select a student A=male B=nursing majorNo- could be both at same time

• 4. Select blood donor A=type O B=female

Are these events mutually exclusive?

• 1. A= roll a 3 on a die B = roll a 4 on a die.Yes – can’t occur at same time

• 2. A= roll a 3 on a die B = roll an even # on dieYes – can’t occur at same time

• 3. Select a student A=male B=nursing majorNo- could be both at same time

• 4. Select blood donor A=type O B=femaleNo – could be both at same time

Are these events mutually exclusive?

• 1. Select a card A=Jack B=Face card

• 2. Select student A=20 yr. B=blue eyes

• 3. Select car A=Ford B=Toyota

Are these events mutually exclusive?

• 1. Select a card A=Jack B=Face cardNo – a jack is BOTH

• 2. Select student A=20 yr. B=blue eyes

• 3. Select car A=Ford B=Toyota

Are these events mutually exclusive?

• 1. Select a card A=Jack B=Face cardNo – a jack is BOTH

• 2. Select student A=20 yr. B=blue eyesNO – could be BOTH 20 and blue eyed

• 3. Select car A=Ford B=Toyota

Are these events mutually exclusive?

• 1. Select a card A=Jack B=Face cardNo – a jack is BOTH

• 2. Select student A=20 yr. B=blue eyesNO – could be BOTH 20 and blue eyed

• 3. Select car A=Ford B=ToyotaYES – can’t be both at same time

Addition Rule: P(A OR B)

• If events A OR B will occur• 1 OR the other, or both!

P(A OR B)= P(A) + P(B) – P(A AND B)Subtracting P(A AND B) avoids double counting

outcomes that occur in BOTH A and B• IF Mutually exclusive:

P(A OR B) = P(A) + P(B)

Examples: Find the Probability

• 1. Select a card. Probability the card is a 4 OR an ace.

• 2. Roll die. Probability rolling a 6 OR an odd.

• 3. Roll a die. Probability rolling a # less than 3 OR odd.

• 4. Select a card. Probability the card a face card OR a heart.

Examples: Find the Probability

• 1. Select a card. Probability the card is a 4 OR an ace. Mutually exclusive

P(4 OR Ace)=P(4)+P(A)=4/52 + 4/52=.154• 2. Roll die. Probability rolling a 6 OR an odd.

• 3. Roll a die. Probability rolling a # less than 3 OR odd.

• 4. Select a card. Probability the card a face card OR a heart.

Examples: Find the Probability

• 1. Select a card. Probability the card is a 4 OR an ace. Mutually exclusive

P(4 OR Ace)=P(4)+P(A)=4/52 + 4/52=.154• 2. Roll die. Probability rolling a 6 OR an odd.

P(6 OR odd)= P(6)+P(odd)=1/6+3/6=.667• 3. Roll a die. Probability rolling a # less than 3 OR odd.

• 4. Select a card. Probability the card a face card OR a heart.

Examples: Find the Probability

• 1. Select a card. Probability the card is a 4 OR an ace. Mutually exclusive

P(4 OR Ace)=P(4)+P(A)=4/52 + 4/52=.154• 2. Roll die. Probability rolling a 6 OR an odd.

P(6 OR odd)= P(6)+P(odd)=1/6+3/6=.667• 3. Roll a die. Probability rolling a # less than 3 OR odd.

P(<3 OR odd)= P(<3)+P(odd) – P(<3 AND odd) 2/6+ 3/6 – (1/6) = 5/6-1/6=4/6=.667

• 4. Select a card. Probability the card a face card OR a heart.

Examples: Find the Probability

• 1. Select a card. Probability the card is a 4 OR an ace. Mutually exclusiveP(4 OR Ace)=P(4)+P(A)=4/52 + 4/52=.154

• 2. Roll die. Probability rolling a 6 OR an odd. P(6 OR odd)= P(6)+P(odd)=1/6+3/6=.667

• 3. Roll a die. Probability rolling a # less than 3 OR odd.P(<3 OR odd)= P(<3)+P(odd) – P(<3 AND odd)

2/6+ 3/6 – (1/6) = 5/6-1/6=4/6=.667• 4. Select a card. Probability the card a face card OR a heart.

P(face OR heart) = P(face)+P(heart)-P(f AND h) 12/52+13/52 –(3/52) = 22/52=.423

Examples:

1. Probability a rep will have sales between $75,000 and $124,000 next month.

2. Probability a rep will have sales between $0 and $49,000 next month.

Sales Months

0-24,999 3

25,000-49,999 5

50,000-74,999 6

75,000-99,999 7

100,000-124,999 9

125,000-149,999 2

150,000-174,999 3

175,000-199,000 1

Examples:

1. Probability a rep will have sales between $75,000 and $124,000 next month. (exclusive)

P(C OR D)=P(C)+P(D) = 7/36+9/36=.4442. Probability a rep will have sales between $0 and

$49,000 next month.

Sales Months

0-24,999 3

25,000-49,999 5

50,000-74,999 6

75,000-99,999 7

100,000-124,999 9

125,000-149,999 2

150,000-174,999 3

175,000-199,000 1

A = sales between 0-$24,999B= sales between $25,000-$49,999C=sales between $75,000-$99,999D=sales between $100,000-$124,999

Total months36

Examples:

1. Probability a rep will have sales between $75,000 and $124,000 next month. (exclusive)

P(C OR D)=P(C)+P(D) = 7/36+9/36=.4442. Probability a rep will have sales between $0 and

$49,000 next month.(exclusive) 3/36+5/36=.222

Sales Months

0-24,999 3

25,000-49,999 5

50,000-74,999 6

75,000-99,999 7

100,000-124,999 9

125,000-149,999 2

150,000-174,999 3

175,000-199,000 1

A = sales between 0-$24,999B= sales between $25,000-$49,999C=sales between $75,000-$99,999D=sales between $100,000-$124,999

Examples: Find the Probability

Rh

• 1. Type O OR A

• 2. Type B OR AB

• 3. Type B OR Rh-negative

• 4. Type O OR Rh-positive

O A B AB Total

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

Blood type

Examples: Find the Probability

Rh

• 1. Type O OR A (exclusive)P(O OR A)= P(O)+P(A)=184/409+164/409=348/409=.85

• 2. Type B OR AB

• 3. Type B OR Rh-negative

• 4. Type O OR Rh-positive

O A B AB Total

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

Blood type

Examples: Find the Probability

Rh

• 1. Type O OR A (exclusive)P(O OR A)= P(O)+P(A)=184/409+164/409=348/409=.85

• 2. Type B OR AB (exclusive) P(B OR AB)=P(B)+P(AB)=45/409+16/409=61/409=.149

• 3. Type B OR Rh-negative

• 4. Type O OR Rh-positive

O A B AB Total

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

Blood type

Examples: Find the Probability

Rh

• 1. Type O OR A (exclusive)P(O OR A)= P(O)+P(A)=184/409+164/409=348/409=.85

• 2. Type B OR AB (exclusive) P(B OR AB)=P(B)+P(AB)=45/409+16/409=61/409=.149

• 3. Type B OR Rh-negativeP(B OR -)=P(B)+P(-)-P(B & -)=45/409+65/409-8/409=.249• 4. Type O OR Rh-positive

O A B AB Total

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

Blood type

Examples: Find the Probability

Rh

• 1. Type O OR A (exclusive)P(O OR A)= P(O)+P(A)=184/409+164/409=348/409=.85

• 2. Type B OR AB (exclusive) P(B OR AB)=P(B)+P(AB)=45/409+16/409=61/409=.149

• 3. Type B OR Rh-negativeP(B OR -)=P(B)+P(-)-P(B & -)=45/409+65/409-8/409=.249

• 4. Type O OR Rh-positiveP(O Or +)=P(O)+P(+)-P(O&+)=184/409+344/409-156/409=.910

O A B AB Total

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

Blood type