<p>Gerri collects characters given away in packets of breakfast cereal. There are four different characters: Horse, Rider, Dog and Bird. Each character is made in four different colours: Orange, Yellow, Green and White. Gerri has collected $56$ items; the numbers of each character and colour are shown in the table.</p>
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<ol class="i-list"><li><div class="list-item-paragraph">Gerri puts all the items in a bag and chooses one item at random.</div><ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that this item is either a Horse or a Rider.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that this item is either a Dog or a Bird but the item is not White.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">Gerri now puts the item back in the bag and chooses two items at random.</div><ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that both of the items are Horses, but neither of the items is Orange.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that Gerri’s two items include exactly one Dog and exactly one item that is Yellow.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Gerri has two favourites among the $16$ possible colour/character combinations. The probability of choosing these two at random from the $56$ items is $\frac{1}{77}.$ Write down all the possibilities for Gerri’s two favourite colour/character combinations.</div><p style="text-align: right;">[3]</p></li></ol>

Gerri collects characters given away in packets of breakfast cereal. There are four different characters: Horse, Rider, Dog and Bird. Each character is made in four different colours: Orange, Yellow, Green and White. Gerri has collected $56$ items; the numbers of each character and colour are shown in the table.

Gerri puts all the items in a bag and chooses one item at random.

Gerri now puts the item back in the bag and chooses two items at random.

Gerri has two favourites among the $16$ possible colour/character combinations. The probability of choosing these two at random from the $56$ items is $\frac{1}{77}.$ Write down all the possibilities for Gerri’s two favourite colour/character combinations.

Probability - General

Probability

International Baccalaureate Higher Level Mathematics

IB HL Math

N2019 P2 Q8

| | Orange | Yellow | Green | White |
|------|--------|--------|-------|-------|
| Horse | 1 | 1 | 3 | 4 |
| Rider | 1 | 1 | 7 | 5 |
| Dog | 3 | 7 | 1 | 6 |
| Bird | 4 | 5 | 6 | 1 |

(i)(a) $\text{P(Horse or Rider)} = \frac{23}{56}$

(i)(b) $\text{P(Dog or Bird} \cap \text{not White)} = \frac{26}{56} = \frac{13}{28}$

(ii)(a) $\text{P(Horse that is not orange)} = \frac{^8C_2}{^{56}C_2} = \frac{28}{1540} = \frac{1}{55}$

or $\frac{8}{56} \times \frac{7}{55} = \frac{1}{55}$

(ii)(b) P(exactly one dog and exactly one yellow)

$= \text{P(Dog not yellow and Yellow not dog)} + \text{P(Yellow dog and other not yellow and not dog)}$

$= \frac{^{10}C_1 \times ^7C_1 + ^7C_1 \times ^{32}C_1}{^{56}C_2}$

$= \frac{70 + 224}{1540} = \frac{294}{1540} = \frac{21}{110}$

(iii) Let the number of favourite characters be $x$ and $y$. Given $x \times y = 20$.

$\text{P(choosing two of them)} = \frac{1}{77}$

$\frac{^xC_1 \times ^yC_1}{^{56}C_2} = \frac{1}{77}$

$x \times y = \frac{1}{77} \times ^{56}C_2 = \frac{1540}{77} = 20$

Possible values: $1 \times 20$, $2 \times 10$, $4 \times 5$

Possibilities: Orange bird (4) and Yellow bird (5), Orange bird (4) and White rider (5), White horse (4) and Yellow bird (5), White horse (4) and White rider (5)

N2019 P2 Q8

N2019 P2 Q8

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