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. Car
2. Goat
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
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 324 times and lost 676 times.

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