Edexcel IAL Mathematics S1.3 probability

$$\frac{q}{q+r}=\frac35$$$$0.13+p+s=\frac{7}{10}$$p+q+r+s+0.12+0.13=1
Solving gives
$$q=0.108,\quad r=0.072.$$

M1 M1 M1 dM1 A1 A1

$$\frac58=0.13+0.12+0.072+s$$s=0.303.

M1 A1

$\mathrm{P}(H \cup W)=\mathrm{P}(H)+\mathrm{P}(W)-\mathrm{P}(H \cap W)$$\mathrm{P}\left(H^{\prime} \cap W\right)=\mathrm{P}(H \cup W)-\mathrm{P}(H)$

M1

$\mathrm{P}(H \cap W)=\frac{3}{8} \times \mathrm{P}(W)$}

$\mathrm{P}\left(H^{\prime} \cap W\right)=\mathrm{P}\left(H^{\prime}\right) \times \mathrm{P}(W)$

M1

$\frac{3}{4}=\frac{3}{8}+\mathrm{P}(W)-\frac{3}{8} \mathrm{P}(W)$}

$\frac{3}{8}=\frac{5}{8} \mathrm{P}(W)$

A1

$\mathrm{P}(W)=\frac{3}{5} *$}

$\mathrm{P}(W)=\frac{3}{5} *$

A1cso*

(4)

$$\mathrm{P}\left(N^{\prime} \mid H\right)=\frac{\frac{3}{8}-\frac{1}{15}}{\frac{3}{8}} \text { or } \frac{\frac{9}{40}+\frac{1}{12}}{\frac{3}{8}} \text { or } 1-\frac{\frac{1}{15}}{\frac{3}{8}}$$$$=\frac{37}{45} \quad=\text { awrt } 0.822$$}}

M1

A1

(2)

B1

M1

M1

M1

A1

(5)

Notes

Total 11

(a)

M1 for use of \mathrm{P(H \cup W)=\mathrm{P}(H)+P(W)-\mathrm{P}(H \cap W) (with at least one value correctly substituted)
or use of $\mathrm{P}\left(H^{\prime} \cap W\right)=\mathrm{P}(H \cup W)-\mathrm{P}(H)$ (with at least one value correctly substituted)}

M1

for use of \mathrm{P(H \cap W)=\mathrm{P}(H) \times \mathrm{P}(W) \quad or use of $\mathrm{P}\left(H^{\prime} \cap W\right)=\mathrm{P}\left(H^{\prime}\right) \times \mathrm{P}(W)$}

A1

a correct equation in \mathrm{P(W) (allow W or x here)}

A1cso*

Correct solution ending with \mathrm{P(W)=\frac{3}{5} with no wrong working seen. Dep. on all previous marks.}

NB

A method which uses \frac{3{5} or $\frac{3}{5} \times \frac{3}{8}\left[=\frac{9}{40}\right]$ can score maximum M1M1A0A0.}

(b)

For \frac{p{\frac{3}{8}} where $0<p<\frac{3}{8}$ use of independence is M0 e.g. $\frac{x \times \frac{3}{8}}{\frac{3}{8}}$}

A1

awrt 0.822

(c)

B1

3 circles labelled. Either N inside H or allow as intersecting circles with $\mathrm{P}\left(N \cap H^{\prime}\right)=0$, but do not allow blank space to be considered 0. Condone missing box for this mark.

Allow all 3 circles overlapping with all zeros correctly labelled.

M1

For \mathrm{P(H \cap W)=\frac{3}{8} \times \frac{3}{5}\left[=\frac{9}{40}\right] seen or correctly placed in Venn diagram.}

M1

For their \frac{3{5}-" \frac{9}{40} "=" \frac{3}{8} (may be implied by the regions in their P(W) adding to 0.6 )}

M1

For their \frac{3{8}-" \frac{9}{40} "-\frac{1}{15}=" \frac{1}{12} "}

A1

Fully correct diagram with \frac{1{4} and box and correct probabilities (allow exact decimal equivalents)}

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Marks