Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Car
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 322 times and lost 678 times.

1. Car
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Car
9. Car
10. Goat
11. Goat
12. Car
13. Car
14. Goat
15. Goat
16. Car
17. Car
18. Car
19. Car
20. Goat
21. Goat
22. Goat
23. Car
24. Goat
25. Goat
26. Goat
27. Goat
28. Car
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Car
51. Goat
52. Goat
53. Goat
54. Car
55. Car
56. Goat
57. Car
58. Car
59. Car
60. Goat
61. Car
62. Goat
63. Goat
64. Goat
65. Goat
66. Goat
67. Goat
68. Car
69. Goat
70. Car
71. Goat
72. Car
73. Goat
74. Car
75. Goat
76. Goat
77. Goat
78. Goat
79. Goat
80. Goat
81. Goat
82. Goat
83. Car
84. Goat
85. Goat
86. Goat
87. Goat
88. Goat
89. Goat
90. Goat
91. Goat
92. Goat
93. Goat
94. Goat
95. Goat
96. Goat
97. Car
98. Goat
99. Car
100. Goat
101. Goat
102. Car
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Car
110. Goat
111. Goat
112. Goat
113. Goat
114. Car
115. Car
116. Goat
117. Goat
118. Goat
119. Goat
120. Car
121. Goat
122. Goat
123. Car
124. Car
125. Goat
126. Car
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Goat
133. Car
134. Goat
135. Goat
136. Car
137. Goat
138. Car
139. Goat
140. Goat
141. Goat
142. Goat
143. Goat
144. Car
145. Car
146. Goat
147. Goat
148. Car
149. Goat
150. Goat
151. Car
152. Goat
153. Goat
154. Goat
155. Goat
156. Car
157. Goat
158. Car
159. Goat
160. Goat
161. Car
162. Goat
163. Car
164. Car
165. Goat
166. Goat
167. Goat
168. Goat
169. Goat
170. Car
171. Goat
172. Car
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Car
179. Goat
180. Car
181. Car
182. Goat
183. Car
184. Goat
185. Goat
186. Car
187. Goat
188. Goat
189. Car
190. Goat
191. Goat
192. Goat
193. Goat
194. Car
195. Goat
196. Car
197. Goat
198. Goat
199. Goat
200. Car
201. Goat
202. Goat
203. Goat
204. Goat
205. Goat
206. Car
207. Car
208. Goat
209. Car
210. Car
211. Car
212. Car
213. Car
214. Car
215. Goat
216. Car
217. Goat
218. Goat
219. Car
220. Goat
221. Goat
222. Goat
223. Goat
224. Goat
225. Goat
226. Car
227. Car
228. Goat
229. Goat
230. Goat
231. Car
232. Car
233. Goat
234. Goat
235. Car
236. Car
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Car
244. Goat
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Goat
251. Goat
252. Goat
253. Car
254. Goat
255. Goat
256. Goat
257. Goat
258. Goat
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Goat
265. Car
266. Car
267. Goat
268. Car
269. Goat
270. Goat
271. Goat
272. Goat
273. Car
274. Car
275. Goat
276. Goat
277. Goat
278. Car
279. Goat
280. Goat
281. Car
282. Car
283. Car
284. Goat
285. Car
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Goat
292. Goat
293. Goat
294. Goat
295. Goat
296. Goat
297. Car
298. Goat
299. Car
300. Goat
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Car
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Car
313. Goat
314. Goat
315. Car
316. Goat
317. Goat
318. Goat
319. Goat
320. Car
321. Goat
322. Car
323. Car
324. Car
325. Car
326. Car
327. Goat
328. Car
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Car
335. Goat
336. Car
337. Goat
338. Goat
339. Goat
340. Car
341. Car
342. Goat
343. Goat
344. Goat
345. Goat
346. Goat
347. Car
348. Goat
349. Goat
350. Goat
351. Car
352. Goat
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Car
365. Goat
366. Goat
367. Goat
368. Goat
369. Goat
370. Goat
371. Car
372. Goat
373. Car
374. Goat
375. Goat
376. Goat
377. Goat
378. Goat
379. Car
380. Goat
381. Goat
382. Goat
383. Car
384. Goat
385. Goat
386. Goat
387. Goat
388. Goat
389. Car
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Car
398. Goat
399. Goat
400. Goat
401. Goat
402. Goat
403. Car
404. Goat
405. Car
406. Goat
407. Car
408. Goat
409. Goat
410. Goat
411. Car
412. Goat
413. Car
414. Car
415. Goat
416. Car
417. Goat
418. Car
419. Car
420. Goat
421. Goat
422. Goat
423. Car
424. Car
425. Car
426. Car
427. Goat
428. Car
429. Car
430. Car
431. Goat
432. Car
433. Goat
434. Goat
435. Goat
436. Goat
437. Goat
438. Goat
439. Car
440. Goat
441. Goat
442. Goat
443. Goat
444. Goat
445. Car
446. Car
447. Goat
448. Car
449. Goat
450. Goat
451. Goat
452. Goat
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Goat
460. Goat
461. Car
462. Goat
463. Car
464. Goat
465. Goat
466. Car
467. Goat
468. Goat
469. Goat
470. Goat
471. Goat
472. Car
473. Car
474. Goat
475. Goat
476. Car
477. Goat
478. Goat
479. Car
480. Car
481. Goat
482. Car
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Car
489. Car
490. Car
491. Goat
492. Car
493. Goat
494. Goat
495. Goat
496. Car
497. Goat
498. Car
499. Car
500. Car
501. Goat
502. Car
503. Car
504. Goat
505. Car
506. Goat
507. Goat
508. Car
509. Goat
510. Goat
511. Car
512. Goat
513. Goat
514. Goat
515. Car
516. Goat
517. Goat
518. Goat
519. Goat
520. Car
521. Car
522. Car
523. Goat
524. Car
525. Goat
526. Goat
527. Car
528. Car
529. Car
530. Car
531. Car
532. Goat
533. Goat
534. Goat
535. Car
536. Car
537. Car
538. Car
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Goat
545. Car
546. Car
547. Goat
548. Goat
549. Goat
550. Car
551. Goat
552. Goat
553. Goat
554. Goat
555. Car
556. Goat
557. Car
558. Goat
559. Goat
560. Goat
561. Car
562. Goat
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Car
570. Goat
571. Goat
572. Goat
573. Goat
574. Goat
575. Goat
576. Car
577. Car
578. Goat
579. Car
580. Goat
581. Car
582. Goat
583. Goat
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Car
591. Goat
592. Car
593. Car
594. Car
595. Car
596. Goat
597. Car
598. Goat
599. Goat
600. Car
601. Goat
602. Goat
603. Goat
604. Car
605. Car
606. Goat
607. Goat
608. Car
609. Car
610. Goat
611. Car
612. Goat
613. Goat
614. Car
615. Car
616. Car
617. Goat
618. Car
619. Car
620. Goat
621. Car
622. Car
623. Goat
624. Goat
625. Goat
626. Car
627. Goat
628. Goat
629. Goat
630. Car
631. Car
632. Car
633. Goat
634. Goat
635. Goat
636. Car
637. Car
638. Goat
639. Goat
640. Goat
641. Car
642. Car
643. Car
644. Goat
645. Car
646. Car
647. Goat
648. Goat
649. Goat
650. Goat
651. Goat
652. Goat
653. Car
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Car
660. Car
661. Goat
662. Goat
663. Car
664. Goat
665. Car
666. Goat
667. Car
668. Goat
669. Goat
670. Car
671. Goat
672. Car
673. Car
674. Goat
675. Car
676. Goat
677. Goat
678. Goat
679. Car
680. Car
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Goat
687. Car
688. Car
689. Goat
690. Goat
691. Goat
692. Goat
693. Goat
694. Car
695. Goat
696. Car
697. Car
698. Car
699. Car
700. Goat
701. Goat
702. Car
703. Car
704. Goat
705. Goat
706. Goat
707. Goat
708. Goat
709. Car
710. Car
711. Goat
712. Car
713. Goat
714. Car
715. Goat
716. Car
717. Goat
718. Goat
719. Car
720. Car
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Goat
727. Goat
728. Car
729. Goat
730. Goat
731. Car
732. Car
733. Car
734. Goat
735. Goat
736. Goat
737. Goat
738. Goat
739. Car
740. Goat
741. Goat
742. Goat
743. Car
744. Goat
745. Goat
746. Car
747. Goat
748. Goat
749. Goat
750. Car
751. Goat
752. Goat
753. Goat
754. Car
755. Goat
756. Goat
757. Car
758. Car
759. Car
760. Goat
761. Goat
762. Goat
763. Car
764. Goat
765. Car
766. Car
767. Goat
768. Goat
769. Car
770. Car
771. Goat
772. Goat
773. Goat
774. Goat
775. Car
776. Goat
777. Goat
778. Goat
779. Goat
780. Car
781. Car
782. Car
783. Goat
784. Goat
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Goat
792. Goat
793. Car
794. Goat
795. Goat
796. Goat
797. Goat
798. Car
799. Goat
800. Goat
801. Goat
802. Car
803. Goat
804. Goat
805. Goat
806. Goat
807. Goat
808. Car
809. Goat
810. Goat
811. Goat
812. Car
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Car
819. Goat
820. Goat
821. Goat
822. Goat
823. Goat
824. Goat
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Car
831. Car
832. Goat
833. Car
834. Car
835. Goat
836. Car
837. Goat
838. Car
839. Goat
840. Goat
841. Goat
842. Goat
843. Goat
844. Car
845. Goat
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Car
852. Goat
853. Car
854. Goat
855. Goat
856. Goat
857. Goat
858. Car
859. Car
860. Goat
861. Car
862. Car
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Car
869. Car
870. Goat
871. Car
872. Goat
873. Goat
874. Car
875. Goat
876. Car
877. Goat
878. Car
879. Goat
880. Goat
881. Goat
882. Goat
883. Goat
884. Goat
885. Goat
886. Car
887. Goat
888. Goat
889. Car
890. Car
891. Car
892. Goat
893. Car
894. Goat
895. Goat
896. Goat
897. Car
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Car
905. Goat
906. Goat
907. Goat
908. Goat
909. Car
910. Goat
911. Goat
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Car
921. Goat
922. Goat
923. Goat
924. Goat
925. Car
926. Goat
927. Goat
928. Goat
929. Car
930. Goat
931. Goat
932. Goat
933. Goat
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Goat
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Goat
947. Car
948. Car
949. Goat
950. Goat
951. Goat
952. Goat
953. Car
954. Goat
955. Goat
956. Goat
957. Car
958. Goat
959. Goat
960. Car
961. Car
962. Goat
963. Car
964. Goat
965. Goat
966. Goat
967. Goat
968. Goat
969. Car
970. Goat
971. Goat
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Car
979. Car
980. Car
981. Goat
982. Car
983. Car
984. Goat
985. Car
986. Goat
987. Goat
988. Goat
989. Goat
990. Car
991. Car
992. Car
993. Goat
994. Goat
995. Car
996. Goat
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 97 times and lost 903 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 36 times and lost 64 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}